Residual Distribution. basics, recents developments, relations with other techniques

Transcription

1 Residual Distribution basics, recents developments, relations with other techniques MARIO RICCHIUTO July 19, 2012

2 INTRODUCTION with historical perspective t u+ F(u) = 0 In the 80 s numerical techniques devised to provide high order monotone approximation to solutions of hyperbolic C.L.s 1. A. Harten (J.Comput.Phys., 1983) : TVD conditions 2. Goodman, LeVeque (Math.Comp.) : TVD in mulitd = first order MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

3 INTRODUCTION with historical perspective t u+ F(u) = 0 In the 80 s numerical techniques devised to provide high order monotone approximation to solutions of hyperbolic C.L.s 1. A. Harten (J.Comput.Phys., 1983) : TVD conditions 2. Goodman, LeVeque (Math.Comp.) : TVD in mulitd = first order This has spawned a number of new approaches Monotonicity conditions 1. ENO/WENO (Harten, Osher, Engquist, Chakravarthy, Shu) 2. TVB conditions (Shu Math.Comp. 1987) 3. Positive coefficient schemes (Spekreijse Math.Comp. 1987) Discretization frameworks 1. Stabilized FE (or central) (Hughes, Morton, Ni, Lerat, Jameson) 2. Discontinuous Galerkin (Cockburn and Shu, starting 1988) 3. Roe s Fluctuation Splitting (starting 1986) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

4 INTRODUCTION with historical perspective t u+ F(u) = 0 Discontinuous Galerkin : smart and elegant combination of existing tools (approximation, Galerkin projection, Riemann solvers, limiters) to generate automatically arbitrary higher order schemes An instant hit... MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

5 INTRODUCTION with historical perspective t u+ F(u) = 0 Fluctuation Splitting/Residual Distribution : A more fundamental and robust approach [...] due to Roe (1986), is that of the genuinely multidimensional upwind schemes. These may be regarded as the true multi-d generalization of 1-D fluctuation splitting [...] These methods are best formulated on simplex-type (finite-element) grids and include newly developed, compact limiters for avoiding oscillations... MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

6 INTRODUCTION with historical perspective t u+ F(u) = 0 Fluctuation Splitting/Residual Distribution : A more fundamental and robust approach [...] due to Roe (1986), is that of the genuinely multidimensional upwind schemes. These may be regarded as the true multi-d generalization of 1-D fluctuation splitting [...] These methods are best formulated on simplex-type (finite-element) grids and include newly developed, compact limiters for avoiding oscillations... B. van Leer, (excerpt from Upwind high resolution methods for compressible flow: from donor cell to residual distribution, Commun.Comput.Phys. 1(2), 2006) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

7 INTRODUCTION with historical perspective t u+ F(u) = 0 Initial idea (P.L.Roe, Num. Meth. Fluid Dyn. 1982) : given values of u on the mesh, integral of F(u) over elements measures the error (fluctuation); decompose the fluctuation in signals allowing to evolve solution values to those solving the problem MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

8 INTRODUCTION with historical perspective t u+ F(u) = 0 Initial idea (P.L.Roe, Num. Meth. Fluid Dyn. 1982) : integral of F(u) over elements measures the error (fluctuation); decompose the fluctuation in signals allowing to evolve solution values to those solving the problem 80 and 90, under the lead of P.L. Roe, H. Deconinck : 1. Genuinely multidimensional upwinding (scalar) 2. Steady state hyperbolic decomposition 3. Each scalar hyperbolic component discretized using MU technique Genuinely multid upwind second order compact (nearest neighbor) second order, nonlinear, positive discretization. Very well adapted to steady supersonic, multidimensional Roe linearization, inexact decompositions in sub-critical case, no unsteady. MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

9 INTRODUCTION with historical perspective t u+ F(u) = 0 Initial idea (P.L.Roe, Num. Meth. Fluid Dyn. 1982) : integral of F(u) over elements measures the error (fluctuation); decompose the fluctuation in signals allowing to evolve solution values to those solving the problem 80 and 90, under the lead of P.L. Roe, H. Deconinck : Genuinely multid upwind second order compact (nearest neighbor) second order, nonlinear, positive discretization. Very well adapted to steady supersonic, multidimensional Roe linearization, inexact decompositions in sub-critical case, no unsteady. 20 years later with contributions of R. Abgrall, T.J. Barth, D. Caraeni, M. Hubbard, C.W. Shu et al. 1. Time dependent problems 2. Conservation without Roe lienarization 3. Higher (than second) accuracy 4. More general data approximation (including discontinuous) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

10 INTRODUCTION with historical perspective t u+ F(u) = 0 Initial idea (P.L.Roe, Num. Meth. Fluid Dyn. 1982) : integral of F(u) over elements measures the error (fluctuation); decompose the fluctuation in signals allowing to evolve solution values to those solving the problem 80 and 90, under the lead of P.L. Roe, H. Deconinck : Genuinely multid upwind second order compact (nearest neighbor) second order, nonlinear, positive discretization. Very well adapted to steady supersonic, multidimensional Roe linearization, inexact decompositions in sub-critical case, no unsteady. 20 years later with contributions of R. Abgrall, T.J. Barth, D. Caraeni, M. Hubbard, C.W. Shu et al. The method has come to a form which is more correctly referred to as a weighted residual method, many of the initial multidimensional upwind fluctuation splitting ideas could not (so far) be retained.. MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

11 INTRODUCTION with historical perspective t u+ F(u) = 0 Initial idea (P.L.Roe, Num. Meth. Fluid Dyn. 1982) : integral of F(u) over elements measures the error (fluctuation); decompose the fluctuation in signals allowing to evolve solution values to those solving the problem 80 and 90, under the lead of P.L. Roe, H. Deconinck 20 years later with contributions of R. Abgrall, T.J. Barth, D. Caraeni, M. Hubbard, C.W. Shu et al. Despite so many contributions the method never really caught up with other approaches based on more sound mathematical foundations (DG), and it definitely has a much lower level of maturity. But some ideas have stuck. Hybrid nature : in between finite element and finite volume. This allows easily to import/export ideas born in this framework to improve others and vice-versa... keeping alive the interest in the method MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

12 INTRODUCTION with historical perspective t u+ F(u) = 0 Initial idea (P.L.Roe, Num. Meth. Fluid Dyn. 1982) : integral of F(u) over elements measures the error (fluctuation); decompose the fluctuation in signals allowing to evolve solution values to those solving the problem Despite so many contributions the method never really caught up with other approaches based on more sound mathematical foundations (DG), and it definitely has a much lower level of maturity What am I going to tell you? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

13 COURSE OUTLINE : Part I 1. Conservative FV discretization and fluctuations 2. Fluctuation splitting/residual distribution framework 3. Design principles 4. Limiters in reverse 5. Relations with other techniques MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

14 COURSE OUTLINE : Part II 1. Higher (than second) orders 2. Time dependent problems 3. Viscous problems 4. Free surface flows 5. Summary, perspectives MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

15 PART I OUTLINE Finite Volume schemes and Fluctuations Design criteria Nonlinear schemes and limiters Relations with other techniques MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

16 1 Conservative Finite Volume schemes and Fluctuations MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

17 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law t u+ x F(u) = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

18 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 x i du i dt + F(u L i+1/2,ur i+1/2 ) F(u L i 1/2,uR i 1/2 ) = 0 u i u i 1 u i+1 i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

19 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 x i du i dt + F(u L i+1/2,ur i+1/2 ) F(u L i 1/2,uR i 1/2 ) = 0 u i 1 u i u i+1 (consistent) F(u,u) = F(u) (L-continuous) F(u,v) F(u,z) K F v z i 1 i i+1 i 1/2 i+1/2 (E-stable. Monotone, etc.) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

20 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : u i t u+ x F(u) = 0 x i du i dt + F i+1/2 F i 1/2 = 0 u i 1 u i+1 i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

21 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 x i du i dt + F i+1/2 F i 1/2 = 0 u i 1 u i u i+1 du i+1 x i+1 dt du i 1 x i 1 dt + F i+3/2 F i+1/2 = 0 + F i 1/2 F i 3/2 = 0 i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

22 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 x i du i dt + F i+1/2 F i 1/2 = 0 u i u i 1 u i+1 i 1 i i+1 i 1/2 i+1/2 du i+1 x i+1 dt du i 1 x i 1 dt + F i+3/2 F i+1/2 = 0 + F i 1/2 F i 3/2 = 0 Conservation from flux cancelation at interfaces MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

23 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 x i du i dt +( F i+1/2 F i )+(F i F i 1/2 ) = 0 u i u i 1 u i+1 i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

24 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law t u+ x F(u) = 0 Conservative FV : φ i+1/2 i φ i 1/2 i {}}{{}}{ du i x i dt + ( F i+1/2 F i )+(F i F i 1/2 ) = 0 u i u i 1 u i+1 i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

25 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 du i x i dt +φi+1/2 i +φ i 1/2 i = 0 u i 1 u i u i+1 at i+1/2 conservation becomes : φ i+1/2 i +φ i+1/2 i+1 = ( F i+1/2 F i )+(F i+1 F i+1/2 ) i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

26 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 du i x i dt +φi+1/2 i +φ i 1/2 i = 0 u i 1 u i u i+1 at i+1/2 conservation becomes : φ i+1/2 i +φ i+1/2 i+1 = F i+1 F i := φ i+1/2 i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

27 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law Conservative FV : t u+ x F(u) = 0 du i x i dt +φi+1/2 i +φ i 1/2 i = 0 u i 1 u i u i+1 at i 1/2 conservation becomes : φ i 1/2 i +φ i 1/2 i 1 = F i F i 1 := φ i 1/2 i 1 i i+1 i 1/2 i+1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

28 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law t u+ x F(u) = 0 Conservative FV : u i du i x i dt +φi+1/2 i +φ i 1/2 i = 0 u i 1 u i+1 i 1 i 1/2 i + 1/2 i i+1 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

29 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law t u+ x F(u) = 0 Conservative FV : du i x i dt +φi+1/2 i +φ i 1/2 i = 0 u i 1 φ i 1/2 u i u i+1 i φ i 1/2 := xf, i 1 (fluctuation) i 1 i 1/2 i + 1/2 i i+1 φ i 1/2 = F i F i 1 (as before...) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

30 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law t u+ x F(u) = 0 Conservative FV : u i 1 φ i 1/2 i u i φ i+1/2 i du i x i dt +φi+1/2 i +φ i 1/2 i = 0 u i+1 i φ i 1/2 := xf, (fluctuation) i 1 φ i 1/2 i = F i F i 1/2 (splitting) i 1 i 1/2 i + 1/2 i i+1 φ i 1/2 i 1 = F i 1/2 F i 1 (splitting) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

31 Finite Volume schemes and Fluctuations in 1D Starting point : conservation law t u+ x F(u) = 0 Conservative FV : u i 1 φ i 1/2 i u i φ i+1/2 i du i x i dt +φi+1/2 i +φ i 1/2 i = 0 u i+1 i i+1 φ i 1/2 := xf, φ i+1/2 := xf i 1 i φ i 1/2 i = F i F i 1/2, φ i+1/2 i = F i+1/2 F i i 1 i 1/2 i + 1/2 i i+1 FV scheme in Fluctuation splitting form... still the same guy though MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

32 Finite Volume schemes and Fluctuations in 2D The multi-d case. Starting point : conservation law t u+ F(u) = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

33 Finite Volume schemes and Fluctuations in 2D l i j K n K il n i = 2( n K ij + nk il ) fij K nk ij C i f ij C j The FV scheme reads C i du i dt + j f ij F ˆndl = 0 i j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

34 Finite Volume schemes and Fluctuations in 2D l i j K n K il n i = 2( n K ij + nk il ) K fij ij C i f ij C j The FV scheme reads C i du i dt + K i K j K fij K i F ˆndl = 0 j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

35 Finite Volume schemes and Fluctuations in 2D l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads C i du i dt + K i K j K i F ij n K ij = 0 K ij n K ij j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

36 Finite Volume schemes and Fluctuations in 2D l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads Discrete conservation C i du i dt + K i K j K i F ij n K ij = 0 F ij n K ij + F ji n K ji = 0 K ij n K ij j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

37 Finite Volume schemes and Fluctuations in 2D l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij Using the identity K C i du i dt + j nk ij = 0 K i K j K i K ij ( F ij F i ) n K ij = 0 n K ij j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

38 Finite Volume schemes and Fluctuations in 2D l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads C i du i dt + K i K j K i K ij ( F ij F i ) n K ij = 0 } {{ } φ K i n K ij j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

39 Finite Volume schemes and Fluctuations in 2D l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads C i du i dt + K i Kφ K i = 0, φ K i = j K i K ij n K ij j ( F ij F i ) n K ij MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

40 Finite Volume schemes and Fluctuations l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads C i du i dt + Discrete conservation K i Kφ K i = 0, φ K i = j K F ij n K ij + F ji n K ji = 0 = j K i K ij n K ij j ( F ij F i ) n K ij φ K j = 1 F i n j := φ K 2 j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

41 Finite Volume schemes and Fluctuations l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads C i du i dt + K i Kφ K i = 0, φ K i = j K i K ij n K ij j ( F ij F i ) n K ij Discrete conservation (F h continuous P 1 finite element approx.) φ K j = φ K = F h j K K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

42 Finite Volume schemes and Fluctuations l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads (F h continuous P 1 finite element approx.) φ K = K F h, i K ij Discrete conservation { }}{ φ K j = φ K j K C i du i dt + K i Kφ K i = 0, φ K i = ( F ij F i ) n K ij j K... but it s still the same guy..!! MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19, n K ij j

43 Finite Volume schemes and Fluctuations l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij The FV scheme reads (F h continuous P 1 finite element approx.) φ K = K F h, i K ij Discrete conservation { }}{ φ K i = φ K j K C i du i dt + K i Kφ K i = 0, φ K i = ( F ij F i ) n K ij j K... but it s still the same guy..!! MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19, n K ij j

44 2 Residual Distribution the framework MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

45 Residual Distribution framework Ω h F(u) = 0 in Ω u = g on Γ λ(u) = u F(u) (1) Γ λ Some notations... Consider Ω h tesselation of Ω Unknowns (Degrees of Freedom, DoF) : u i u(m i ) M i Ω h a given set of nodes (vertices +other dofs) u h : continuous polynomial interpolation u h = i ψ i u i MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

46 Residual Distribution framework 1. K Ω h compute : φ K = K F h (u h ) K φ K i i K 2. Distribution : φ K = Distribution coeff.s : φ K i =β K i φ K φ K φ K 1 φ K 2 φ K 3 3. Compute nodal values : solve algebraic system T i T φ K i = 0, i Ω h (2) i MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

47 Residual Distribution framework Seek the steady limit of C i du i dt + K i K φ K i t K i K φ K i = 0 (3) The idea of Residual Distribution or Fluctuation Splitting Fluctuations & Signals (Roe, Num.Meth.Fluid Dyn., 1982) From an initial guess, nodal values evolve to steady state due to signals proportional to cell residuals (Roe s fluctuation) 1 - Compute fluctuation 2 - Split 3 - Gather signals K φ K φ K 1 φ K 2 φ K 3 i 4 - Evolve eq. ((3)) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

48 Structural conditions Conservation. LW theorem : convergence (if...) to weak solution? Stability. which form of stability (energy/entropy, equivalent algebraic condition, convergence?), choice of φ K i Accuracy. characterization of the error, choice of φ K i Oscillations. monotonicity preserving schemes, choice of φ K i MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

49 Conservation Conservation, LW theorem for RD and F(u) = 0 Under some (standard) continuity assumptions on φ K and φ K i the discrete solution u h converges (if...!) to a weak solution of the continuous problem, provided that (Abgral, Barth SISC, 2002 ; Abgrall, Roe J.Sci.Comp., 2003) : φ K j (u h) = φ K (u h ) = j K K F h (u h ) ˆndl for some continuous approximation of the flux F h. MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

50 Conservation Conservation, LW theorem for RD and F(u) = 0 φ K j (u h) = φ K (u h ) = F h (u h ) ˆndl j K for some continuous approximation of the flux F h. K In practice, approach 1 Set F h = j K ψ jf j and integrate exactly. This gives the P 1 element residual seen for the FV scheme MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

51 Conservation Conservation, LW theorem for RD and F(u) = 0 φ K j (u h ) = φ K (u h ) = F h (u h ) ˆndl j K for some continuous approximation of the flux F h. In practice, approach 2 Set F h = F(v h ) with v some set of variables, v h = j ψ jv j, apply Gauss Formulae on K (Csik, Ricchiuto, Deconinck, J.Comput.Phys, 2002) K φ K (u h ) = f K f F h (x) ˆndl = G p f ω q F h (x q ) ˆn f f K q=1 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

52 Conservation Conservation, LW theorem for RD and F(u) = 0 φ K j (u h ) = φ K (u h ) = F h (u h ) ˆndl j K for some continuous approximation of the flux F h. In practice, approach 3 Set F h = F(v h ) with v some set of variables, v h = j ψ jv j, and integrate exactly of below truncation error (Deconinck, Struijs, Roe K Computers & Fluids, 1993 ; Abgrall, Barth SISC, 2002) φ K = K F v (v h) v h dk MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

53 Conservation Conservation, LW theorem for RD and F(u) = 0 φ K j (u h) = φ K (u h ) = F h (u h ) ˆndl j K for some continuous approximation of the flux F h. In practice Schemes are more often written so that we recover at the end φ K (u h ) = f K f K F h (x) ˆndl = f K G p f ω q F h (x q ) ˆn f q=1 for some (edge) continuous polynomial reconstruction F h (x) which remains one of the degrees of freedom of the method MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

54 Conservation Conservation, LW theorem for RD and F(u) = 0 φ K j (u h) = φ K (u h ) = F h (u h ) ˆndl j K for some continuous approximation of the flux F h. This only guarantees that if discontinuous solutions are approximated, the correct jump (Rankine-Hugoniot) conditions are recovered K What about stability, accuracy, etc.? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

55 3 DESIGN CRITERIA Accuracy, stability, and all that jazz MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

56 Design criteria 1. K Ω h compute : φ K = F h (u h ) K φ K i i K 2. Distribution : φ K = Distribution coeff.s : φ K i =β K i φ K K φ K φ K 1 3. Compute nodal values : solve algebraic system C i du i dt + T i T φ K i = 0, t (4) φ K 2 φ K 3 i MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

57 What can we say about the stability of this method? First : what is stability? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

58 What can we say about the stability of this method? First : what is stability? Assume, for h fixed you do u n+1 i = u n i ω i K i K φ K i (u n h ), ω i = t C i MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

59 What can we say about the stability of this method? First : what is stability? More abstractly (ω a scalar,e.g. ω = min i ω i ) for h fixed you do u n+1 = u n ω(a h u n f) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

60 What can we say about the stability of this method? First : what is stability? More abstractly (ω a scalar,e.g. ω = min i ω i ) for h fixed you do u n+1 = u n ω(a h u n f) A condition for convergence with n but h fixed (I ωa h )u 2 r u 2, u and with r < 1 which is equivalent to u t A h u 1 r 2ω u 2 + ω 2 A hu 2 C h u 2 0 u MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

61 What can we say about the stability of this method? First : what is stability? More abstractly (ω a scalar,e.g. ω = min i ω i ) for h fixed you do u n+1 = u n ω(a h u n f) A condition for convergence with n but h fixed (I ωa h )u 2 r u 2, u and with r < 1 which is equivalent to u t A h u 1 r 2ω u 2 + ω 2 A hu 2 C h u 2 0 u Coercivity... MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

62 Stability and energy Consider the steady limit of t u+ a u = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

63 Stability and energy Consider the steady limit of Semi-discrete counterpart t u+ a u = 0 C i du i dt + K i K φ K i = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

64 Stability and energy Consider the steady limit of Semi-discrete counterpart t u+ a u = 0 C i du i dt + K i K φ K i = 0 Energy budget The equivalent of the quantity u t A h u seen in the previous slides is u t A h u u i φ K i i Ω h K i K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

65 Stability and energy Consider the steady limit of Semi-discrete counterpart t u+ a u = 0 C i du i dt + K i K φ K i = 0 Energy budget The equivalent of the quantity u t A h u seen in the previous slides is u t A h u u i φ K i K Ω h i K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

66 Stability and energy Consider the steady limit of Semi-discrete counterpart t u+ a u = 0 C i du i dt + K i K φ K i = 0 Energy budget The equivalent of the quantity u t A h u seen in the previous slides is u t A h u u i φ K i K Ω h i K }{{} φ E K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

67 Stability and energy Consider the steady limit of Semi-discrete counterpart t u+ a u = 0 C i du i dt + K i K φ K i = 0 Energy budget The equivalent of the quantity u t A h u seen in the previous slides is u t A h u K Ω h φ E K What is φ E K? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

68 Stability and energy Starting from Energy budget C i du i dt + K i K du i C i u i dt + i Ω h φ K i = 0 K Ω h φ E K = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

69 Stability and energy Starting from C i du i dt + K i K φ K i = 0 Energy budget i Ω h C i d dt ( ) u 2 i + φ E K 2 = 0 K Ω h MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

70 Stability and energy Starting from Energy budget C i du i dt + Ω h with the energy density and with E h = K i K de h dt + φ K i = 0 K Ω h φ E K = 0 E = u2 2 i Ω h E i ψ i (piecewise linear) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

71 Stability and energy Saying that 0 u t A h u is equivalent to K Ω h φ E K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

72 Stability and energy Saying that 0 < u t A h u K Ω h φ E K is equivalent to Energy stability Ω h de h dt = K Ω h φ E K 0 with the energy density and with E h = E = u2 2 i Ω h E i ψ i (piecewise linear) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

73 Stability and energy Saying that is equivalent to 0 < u t A h u K Ω h φ E K Energy stability (modulo boundary conditions) Ω h de h dt = Ω h E h a ˆndl δ E, δ E 0 what one would like is to find that φ E K = E h a ˆndl+δ K E, δe K 0 K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

74 Stability and upwinding Consider again the steady limit of t u+ a u = 0 A geometrical view of advection... MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

75 Stability and upwinding Consider again the steady limit of t u+ a u = 0 A geometrical view of advection... 1-target triangle The inlet region is an edge 1 node downstream : 1 target k j = a n j 2 > 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

76 Stability and upwinding Consider again the steady limit of t u+ a u = 0 A geometrical view of advection... 1-target triangle The inlet region is an edge 1 node downstream : 1 target k j = a n j 2 > 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

77 Stability and upwinding Consider again the steady limit of t u+ a u = 0 A geometrical view of advection... 2-target triangle The outlet region is an edge 2 nodes downstream : 2 targets k j = a n j 2 > 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

78 Stability and upwinding Consider again the steady limit of t u+ a u = 0 A geometrical view of advection... 2-target triangle The outlet region is an edge 2 nodes downstream : 2 targets k j = a n j 2 > 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

79 Stability and upwinding Consider now the semi-discrete RD advection equation : C i du i dt + K i K A geometrical view of advection... φ K i = 0 Multidimensional Upwinding (MU) Multidimensional Upwind (MU) schemes only split φ K to downstream nodes, i.e. those for which k j > 0. All MU schemes reduce to the same in the 1-target case. MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

80 Stability and upwinding Consider now the semi-discrete RD advection equation : C i du i dt + K i K A geometrical view of advection... φ K i = 0 Multidimensional Upwinding (MU) Multidimensional Upwind (MU) schemes In 1-target elements if k 1 > 0 (node 1 only node downstream) φ K 1 = φk, φ K 2 = φk 3 = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

81 Stability and upwinding Consider now the semi-discrete RD advection equation : C i du i dt + K i K A geometrical view of advection... φ K i = 0 Multidimensional Upwinding (MU) Multidimensional Upwind (MU) schemes In 2-targets elements if k 1 < 0 (node 1 only node upstream) φ K 1 = 0, φk 2 +φk 3 = φk MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

82 Stability and MU schemes Example 1 : Roe s optimal N scheme 1 2 φ K ( a 1) = K a1 u h 2 φ K ( a 2) = K a2 u h a 1 a φ N 1 = φ K φ N 2 = φ N 3 = 0 a a 1 φ N 3 = 0 φ N 1 = φ K ( a 1) φ N 2 = φ K ( a 2) The formula (Roe Cranfield U.Tech.Rep., 1987 ; Roe, Sidilkover SINUM, 1992) kj u j φ N i = k + i (u i u in ), u in = j K kj j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

83 Stability and MU schemes Example 2 : the LDA scheme The LDA scheme reads 2 φ LDA i (u h ) = βi LDA φ K (u h ) K 3 where β LDA i = k+ i k j + j K a K 3 / K K 2 / K K 2 1 recalling that for the advection equation (u h piecewise linear) φ K (u h ) = a u h φ K 1 K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

84 Stability and MU The following properties can be easily shown : 1. MU schemes, 1-target (Deconinck, Ricchiuto Enc.Comput.Mech., 2007) φ E K = K E h a ˆndl +δ E K, δe K 0 2. N scheme energy stable (Barth, NASA 1996 ; Abgrall, Barth SISC, 2002) 3. LDA scheme, 2-targets (Deconinck, Ricchiuto Enc.Comput.Mech., 2007) φ E LDA = ( k + ) ( u 2 ) out j 2 u2 in + δ 2 LDA, E δlda E 0 j K }{{} NRG balance along streamline Multidimensional upwinding does the job... MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

85 Stability, upwinding, and dissipation 1. FV scheme (1st order upwind) NRG stable (Barth, NASA 1996 ; Abgrall, Barth SISC, 2002), also E-flux schemes by (Osher SINUM, 1984) 2. Streamline upwind finite element scheme SUPG, (Hughes, Brooks CMAME, 1982) : MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

86 Stability, upwinding, and dissipation 1. FV scheme (1st order upwind) NRG stable (Barth, NASA 1996 ; Abgrall, Barth SISC, 2002), also E-flux schemes by (Osher SINUM, 1984) 2. Streamline upwind finite element scheme (SUPG) (Hughes, Brooks CMAME, 1982) : ψ i F h (u h )+ a(u h ) ψ i τ a(u h ) u h = 0 Ω K Ω K h h can be written as the RD scheme φ K i = 0 with φ K i = K K i K ψ i F h (u h )+ a(u h ) ψ i τ a(u h ) u h K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

87 Stability, upwinding, and dissipation 1. FV scheme (1st order upwind) NRG stable (Barth, NASA 1996 ; Abgrall, Barth SISC, 2002), also E-flux schemes by (Osher SINUM, 1984) 2. Streamline upwind finite element scheme SUPG : φ SUPG i = ψ i a u h + a ψ i τ a u h K one easily checks that since j ψ j = 1 and j ψ j = 0 φ SUPG j = a u h = φ K (u h ) j K K K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

88 Stability, upwinding, and dissipation 1. FV scheme (1st order upwind) NRG stable (Barth, NASA 1996 ; Abgrall, Barth SISC, 2002), also E-flux schemes by (Osher SINUM, 1984) 2. Streamline upwind finite element scheme (SUPG) : φ E SUPG = K u 2 h 2 a ˆndl+ K a u h τ a u h } {{ } Streamline dissipation 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

89 Stability, upwinding, and dissipation 1. FV scheme (1st order upwind) NRG stable (Barth, NASA 1996 ; Abgrall, Barth SISC, 2002), also E-flux schemes by (Osher SINUM, 1984) 2. Streamline upwind finite element scheme (SUPG) : φ E SUPG = K u 2 h 2 a ˆndl+ K a u h τ a u h } {{ } Streamline dissipation 0 3. Lax-Friedrich s/rusanov scheme φ LF i = ψ i a u h +α LF (u i u j ) K j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

90 Stability, upwinding, and dissipation 1. FV scheme (1st order upwind) NRG stable (Barth, NASA 1996 ; Abgrall, Barth SISC, 2002), also E-flux schemes by (Osher SINUM, 1984) 2. Streamline upwind finite element scheme (SUPG) : φ E SUPG = K u 2 h 2 a ˆndl+ 3. Lax-Friedrich s/rusanov scheme φ E LF = K K u 2 h 2 a ˆndl + α LF 3 a u h τ a u h } {{ } Streamline dissipation 0 (u i u j ) 2 i,j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

91 Stability, upwinding, and dissipation Upwinding has beneficial effect in terms of energy stability MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

92 Design criteria : what is the truncation error? By Taylor expansion : no way (unless meshes with particular structure are considered) Error analysis based on variational form : 1. which variational form? 2. NRG stability not enough, no coercivity no tools for analysis Idea : use weak form to define error (consistency estimate) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

93 Design criteria : what is the truncation error? Idea : use weak form to define an integral truncation error ϕ F(u)dx+BCs = 0 Ω Ω ϕ F h (u h )dx+bcs = ε h with u a smooth exact (classical) solution This gives a consistency estimate.. What is ε h? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

94 Design criteria, consistency analysis What do we have...? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

95 Design criteria, consistency analysis What do we have...? Consider 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) with w h a continuous polynomial approximation of degree k (e.g standard Lagrange elements) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

96 Design criteria, consistency analysis Continuous Lagrange elements P 1 P 2 P 3 Q 1 Q 2 Q 3 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

97 Design criteria, consistency analysis What do we do...? Consider 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) Take the steady RD scheme K i K approximating F in node i φ K i (u h) = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

98 Design criteria, consistency analysis What do we do...? Consider 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) Formally replace the nodal values of u h, computed by the scheme, with those of the exact solution w, exaclty as done in finite difference TE analysis MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

99 Design criteria, consistency analysis What do we do...? Consider 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) We obtain K i K φ K i (w h ) 0 since of course the nodal values of the exact solution w do not verify the discrete equations MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

100 Design criteria, consistency analysis What do we do...? Consider 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) Given ϕ a C0 r (Ω) class function, r large enough, define ǫ h := φ K i (w h ) ϕ i i Ω h K i K A global measure of how much the discrete equations differ from the continuous one MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

101 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) ǫ h = K Ω h i K ϕ i φ K i (w h) = ǫ a +ǫ d ǫ a = ϕ h (F h (w h ) F(w)) Ω } h {{} approximation error ϕ h = ψ j ϕ j K Ω h j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

102 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) ǫ h = K Ω h i K ϕ i φ K i (w h) = ǫ a +ǫ d ǫ d = K Ω h i,j K ϕ i ϕ j (φ K i (w h ) φ G i (w h )) n K DoF } {{ } distribution error φ G i (w h) = ψ i (F h (w h ) F(w)) (Galerkin proj.) K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

103 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

104 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) We get easily the estimates ǫ a = ϕ h (F h (w h ) F(w)) C ah k+1 Ω h MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

105 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) We get easily the estimates φ G (w h ) = K ψ i (F h (w h ) F(w)) C a hk+2 ϕ i ϕ j h ϕ Ch MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

106 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) We get easily the estimates ǫ d = ϕ i ϕ j K Ω h i,j K n K (φ K i (w h) φ G i (w h)) DoF C Ωh h 2 Ch (sup K sup i K φ K i (w h) +C a hk+2 ) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

107 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) in L 2 from approximation theory, see e.g. (Ern, Guermond Springer, 2004) We get easily the estimates ǫ d C h 1 sup K sup φ K i (w h ) +C a h k+1 i K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

108 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) ǫ h C a h k+1 +C h 1 sup K sup φ K i (w h ) i K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

109 Design criteria, consistency analysis What do we do...? Estimate ǫ h (Abgrall, Roe J.Sci.Comp., 2003 ; Ricchiuto, Abgrall, Deconinck J.Comput.Phys, 2007) ǫ h C a h k+1 +C h 1 sup K sup φ K i (w h ) i K For a polynomial approximation of degree k, a sufficient condition to have a ǫ h Ch k+1 is (in 2d) φ K i (w h ) = O(h k+2 ), K h, i K A local TE condition.. MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

110 Design criteria, high order schemes For a polynomial approximation of degree k, a sufficient condition to have a ǫ h Ch k+1 is (in 2d) φ K i (w h ) = O(h k+2 ), K h, i K 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

111 Design criteria, high order schemes For a polynomial approximation of degree k, a sufficient condition to have a ǫ h Ch k+1 is (in 2d) φ K i (w h ) = O(h k+2 ), K h, i K 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) High order prototype 1, Petrov-Galerkin φ K i (u h) = ωi K F h(u h ), ωi K < C < K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

112 Design criteria, high order schemes For a polynomial approximation of degree k, a sufficient condition to have a ǫ h Ch k+1 is (in 2d) φ K i (w h ) = O(h k+2 ), K h, i K 1. w H k+1 smooth solution : F(w) = u F(w) w = 0 2. w w h = O(h k+1 ), F(w) F h (w h ) = O(h k+1 ) 3. (w w h ) = O(h k ), (F(w) F h (w h )) = O(h k ) High order prototype 2, accuracy preserving RD φ K i (u h ) = βi K F h (u h ) = βi K φ K (u h ), βi K < C < K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

113 High order schemes, examples LDA scheme (P 1 ) elements β LDA i = k + i ( j K k + j ) 1 SUPG ω K i = ψ i + a(u h ) ψ i τ, a(u h ) = u F(u h ) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

114 4 LIMITERS using them in reverse MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

115 Nonlinear high order schemes So far we have 1. A stability criterion requiring an upwind bias (other stabilization strategies mentioned later if time..) 2. An accuracy (consistency) criterion requiring bounded weights in the residual splitting How about discontinuity capturing? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

116 Discontinuity capturing : positivity C i du i dt = K i K Positive coefficient scheme (Spekreijse, Math.Comp. 49, 1987) A scheme for which φ K i = c K ij(u i u j ) with c K ik 0 j K j i s said to be LED (Local Extremum Diminishing) φ K i MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

117 Discontinuity capturing : positivity C i un+1 i t u n i = K i K c K ij (un i un j ) j K j i Positive coefficient scheme (Spekreijse, Math.Comp. 49, 1987) When combined with Explicit Euler time integration (or with another boundedness preserving time integration scheme) LED leads to c ij u n j u n+1 i = j where c ij 0, j c K ij = 1 provided t C i c ij 1 j In this case the scheme is said (by abuse of language) to be positive MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

118 Discontinuity capturing : positivity C i un+1 i t u n i = K i K c K ij (un i un j ) j K j i Positive coefficient scheme (Spekreijse, Math.Comp. 49, 1987) When combined with Explicit Euler time integration (or with another boundedness preserving time integration scheme) LED leads to c ij u n j u n+1 i = j where c ij 0, j c K ij = 1 provided t C i c ij 1 j In this case the scheme is said (by abuse of language) to be positive MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

119 Discontinuity capturing : positivity u n+1 i = j c ij u n j with c ij 0, c K ij = 1 j Positive coefficient scheme (Spekreijse, Math.Comp. 49, 1987) A positive scheme verifies the discrete max principle minu n j un+1 i j maxu n j j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

120 Positive schemes : examples Example 1 : Roe s optimal N scheme 1 2 φ K ( a 1) = K a1 u h 2 φ K ( a 2) = K a2 u h a 1 a φ N 1 = φ K φ N 2 = φ N 3 = 0 a a 1 φ N 3 = 0 φ N 1 = φ K ( a 1) φ N 2 = φ K ( a 2) The formula (Roe Cranfield U.Tech.Rep., 1987 ; Roe, Sidilkover SINUM, 1992) kj u j φ N i = k + i (u i u in ), u in = j K kj j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

121 Nonlinear high order schemes Example 2 : Lax-Friedrich s distribution 2 φ T 1 LF φi = K ψ i F h +α LF (u i u j ) j K for positivity (scalar case) 3 α LF h K sup u F(u h (x)) x K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

122 Nonlinear high order schemes Bad news... (Godunov) All linear positive (LED) schemes are first order accurate... MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

123 Nonlinear high order schemes Where does the limiter come in Recall that one prototype of a high order scheme is φ K i (u h) = β K i φ K (u h ), β K i C < MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

124 Nonlinear high order schemes Where does the limiter come in Recall that one prototype of a high order scheme is φ K i (u h) = β K i φ K (u h ), β K i C < For linear positive coefficient schemes φ P i (u h) = j Kc K ij (u i u j ), c K ij 0 Formally we have c K ij (u i u j ) β P i (u h) = j K φ K (u h ) in general unbounded! MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

125 Nonlinear high order schemes Where does the limiter come in The idea : apply a limiter function to bound the distribution coefficient MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

126 Nonlinear high order schemes Where does the limiter come in The idea : apply a limiter function to bound the distribution coefficient βi LP (u h ) = ψ( βi P(u h) ) ψ ( βj P(u h) ) j K The scaling on the denominator guarantees that j β LP i = 1 What are the conditions on the limiter function ψ( )? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

127 Nonlinear high order schemes Where does the limiter come in Linear positive coefficient schemes φ P i (u h) = j K β LP c K ij (u i u j ),c K ij 0 ; βp i (u h) = φp i (u h) φ K (u h ) unbounded i (u h ) = ψ( βi P(u h) ) ψ ( βj P(u ) limited distribution coefficient h) j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

128 Nonlinear high order schemes Where does the limiter come in Linear positive coefficient schemes φ P i (u h ) = j K β LP c K ij(u i u j ),c K ij 0 ; β P i (u h ) = φp i (u h) φ K (u h ) unbounded i (u h ) = ψ( βi P(u h) ) ψ ( βj P(u ) limited distribution coefficient h) j K φ LP i (u h ) = βi LP φ K = βlp i β P i Provided ψ(r) 0 and ψ(r) r }{{} γ P i 0 φ P i = j K 0 we have c LP ij (u i u j ), c LP ij = γ P i c K ij 0! MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

129 High order schemes High order RD scheme 1. Compute cell residual φ K = K F h (u h ) dl 2. Compute linear positive distribution φ P i = j ck ij (u i u j ) 3. Limit β P i = φ P i /φk β LP i = ψ(β P i )/( j ψ(βp j )) 4. Distribute cell residual φ K i = β LP i φ K 5. Evolve C i du i dt = K i K φ K i until steady state MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

130 High order schemes High order RD scheme 1. Compute cell residual φ K = K F h (u h ) dl 2. Compute linear positive distribution φ P i = j ck ij (u i u j ) 3. Limit β P i = φ P i /φk β LP i = ψ(β P i )/( j ψ(βp j )) 4. Distribute cell residual φ K i = β LP i φ K 5. Evolve C i du i dt = K i K φ K i until steady state The simplest possible choice for ψ( ) is ψ(r) = max(0, r) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

131 Examples Rotational advection Scalar example : a u = 0 with a = (y,1 x) and bcs { cos(2π(x+0.5)) 2 if x [ 0.75, 0.25] u in = 0 otherwise y x inlet outlet MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

132 y y Examples (cont d) Rotational advection N and Limited N (LN) schemes N scheme 1 N LN Exact 0.8 outlet data x u LN scheme x x MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

133 y y Examples (cont d) Rotational advection LF and Limited LF (LLF) schemes LF scheme 1 LF LLF Exact 0.8 outlet data x u LLF scheme x x MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

134 Examples (cont d) Burger s equation Scalar example : F(u) = 0 with F(u) = (u, u2 2 ) and bcs u(x,y = 0) = 1.5 2x u in MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

135 y y Examples (cont d) Burger s equation N scheme N and Limited N (LN) schemes y = x 1 LN scheme u y = N LN x x MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

136 y y Examples (cont d) Burger s equation LF scheme LF and Limited LF (LLF) schemes y = x LLF scheme u LF LLF y = x x MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

137 Remarks on extension to systems Historical perspective Two approaches (Roe J.Comput.Phys, 1986 ; Nishikawa, Rad, Roe AIAA Conf. 2001) and (van der Weide, Deconinck Comput.Fluid Dyn., Wiley 1996) 1. Local projection (wave decomposition) of the continuous PDE to obtain (possibly decoupled) scalar equations discretized independently 2. Formal matrix generalization in which the scalar flux vector is replaced by a tensor and the k i = a n i /2 coefficients become matrix flux Jacobians MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

138 Remarks on extension to systems Practical implementation Hybrid of the two (Abgrall, Mezine J.Comput.Phys, 2004 ; Ricchiuto, Csik, Deconinck J.Comput.Phys, 2005) : Matrix formulation for linear first order schemes Projection onto characteristic directions to obtain scalar residuals to work with for the limiting procedure (similar to FV limiting on characteristic var.s) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

139 Example 1 : Mach 3.6 scramjet inlet (Euler, perfect gas) Mesh N scheme Nonlinear scheme MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

140 Example 2 : Mach 10 bow shock (Euler, perfect gas) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

141 6 ON THE RELATIONS WITH FEM, FV, DG, WENO FD, etc. etc. MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

142 Relations with other techniques Continuous, Stabilized Finite Elements By nature of the underlying approximation, these methods bear close resemblance to stabilized continuous Galerkin methods as e.g. the SUPG of (Hughes, Brooks CMAME, 1982) (bcs omitted) ψ i F h (u h )+ a(u h ) ψ i τ a(u h ) u h = 0 Ω K Ω K h h which, as seen, can be written as the RD scheme φ K i = 0 with φ K i = ψ i F h (u h )+ a(u h ) ψ i τ a(u h ) u h K i K K K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

143 Relations with other techniques Continuous, Stabilized Finite Elements More generally, a Petrov-Galerkin method with test space spanned by functions {ω i } i Ωh such that K Ω h i K = j Kω ωi K = 1 j K can be recast as a RD scheme φ K i = 0 with φ K i = K i K K ω K i F h(u h ) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

144 Relations with other techniques : RD and FV Which numerical flux defines your conservative statement? The first relation we have already seen : FV schemes can be written such that they sit in an element l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij i K ij n K ij j C i du i dt + K i Kφ K i = 0, φ K i = ( F ij F i ) n K ij j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

145 Relations with other techniques : RD and FV Which numerical flux defines your conservative statement? The first relation we have already seen : FV schemes can be written such that they sit in an element Here conservation is expressed by the FV flux function F C i du i dt + K i Kφ K i = 0, φ K i = ( F ij F i ) n K ij j K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

146 Relations with other techniques : RD and FV Which numerical flux defines your conservative statement? A different view is to recast RD as FV on the median dual l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij C i du i dt + K i K j K i F ij n K ij = 0 K ij n K ij j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

147 Relations with other techniques : RD and FV Which numerical flux defines your conservative statement? A different view is to recast RD as FV on the median dual l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij C i du i dt = K i K j K F ij n K ij i K ij n K ij What definition of F ij such F ij n K ij = βk i φk j K for a given φ K i = β K i φ K j MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

148 Relations with other techniques : RD and FV Which numerical flux defines your conservative statement? A different view is to recast RD as FV on the median dual l i j K n K il n i = 2( n K ij + nk il ) f C i C j f ij i K ij n K ij j C i du i dt = K i K j K F ij n K ij F RD ij What numerical flux defines local conservation on the median dual cell for RD? MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

149 Relations with other techniques : RD and FV Which numerical flux defines your conservative statement? A different view is to recast RD as FV on the median dual. C i du i dt = j K K i K j K F RD ij n K ij F RD ij n K ij = βk i ΦK, i F RD ij What numerical flux defines local conservation on the median dual cell for RD? Answer in (Abgrall, 2012) F RD ij n ij = Ψ i Ψ j with Ψ i = β K i φ K F i n i 2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

150 Relations with other techniques : RD and FV Which numerical flux defines your conservative statement? A different view is to recast RD as FV on the median dual. C i du i dt = K i K j K F RD ij n K ij F RD ij What numerical flux defines local conservation on the median dual cell for RD? Answer in (Abgrall, 2012) F RD ij n ij = Ψ i Ψ j with Ψ i = β K i φ K F i n i 2 Consistent 3-states numerical flux function This is still the RD scheme we start with MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

151 Relations with other techniques : RD and FV u i u i 1 u i+1 i 1 i i+1 i 1/2 i+1/2 FV fluxes from RD schemes Starting point : conservation law Conservative FV : t u+ x F(u) = 0 x i du i dt + F i+1/2 F i 1/2 = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

152 Relations with other techniques : RD and FV u i (x) u i 1 (x) u i+1 (x) F i+1/2 = F(u i(x i+1/2 ),u i+1(x i+1/2 )) F i 1/2 = F(u i 1(x i 1/2 ),u i(x i 1/2 )) i 1 i 1/2 i i+1 i+1/2 FV fluxes from RD schemes Conservative FV : x i du i dt + F i+1/2 F i 1/2 = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

153 Relations with other techniques : RD and FV u i 1 (x) u i (x) u i+1 (x) F i = F(u i(x i+1/2 ) F(u i(x i 1/2 )) φ i+1/2 i = F i+1/2 F(u i(x i+1/2 )) i 1 i 1/2 i i+1 i+1/2 φ i+1/2 i FV fluxes from RD schemes Conservative FV, reformulation : = F(u i+1(x i+1/2 )) F i+1/2 du i x i dt + F i +φ i+1/2 i +φ i 1/2 i = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

154 Relations with other techniques : RD and FV u i 1 (x) u i (x) u i+1 (x) F i = F(u i(x i+1/2 ) F(u i(x i 1/2 )) φ i+1/2 i = F i+1/2 F(u i(x i+1/2 )) i 1 i 1/2 i i+1 i+1/2 φ i+1/2 i FV fluxes from RD schemes Conservative FV, reformulation : = F(u i+1(x i+1/2 )) F i+1/2 du i x i dt + F i +φ i+1/2 i +φ i 1/2 i = 0 x i+1 du i+1 dt + F i+1 +φ i+3/2 i+1 +φ i+1/2 i+1 = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

155 Relations with other techniques : RD and FV u i 1 (x) u i (x) u i+1 (x) F i = F(u i(x i+1/2 ) F(u i(x i 1/2 )) φ i+1/2 i = F i+1/2 F(u i(x i+1/2 )) i 1 i 1/2 i i+1 i+1/2 φ i+1/2 i FV fluxes from RD schemes Conservative FV, reformulation : φ i+1/2 i = F(u i+1(x i+1/2 )) F i+1/2 +φ i+1/2 i+1 := φ i+1/2 = F(u i+1 (x i+1/2 )) F(u i (x i+1/2 )) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

156 Relations with other techniques : RD and FV u i (x) u i 1 (x) h l u i+1 (x) i x x + i 1/2 i 1/2 FV fluxes from RD schemes x i+1/2 x + i+1/2 Integrate over each control volume : no need for numerical flux (continuity through ghost elements) In each ghost element apply any RD scheme which will provide a definition for the numerical flux as x + i±1/2 x i±1/2 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

157 Relations with other techniques : RD and FV u i (x) u i 1 (x) h l u i+1 (x) i x x + i 1/2 i 1/2 FV fluxes from RD schemes x i+1/2 x + i+1/2 u n+1 i x i t u n i + F i +β i+1/2 i φ i+1/2 +β i 1/2 i φ i+1/2 = 0 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

158 Relations with other techniques : RD and FV u i (x) u i 1 (x) u i+1 (x) F i+1/2 = F(u i(x i+1/2 ),u i+1(x i+1/2 )) F i 1/2 = F(u i 1(x i 1/2 ),u i(x i 1/2 )) i 1 i 1/2 i i+1 i+1/2 FV fluxes from RD schemes u n+1 i x i t F RD u n i + F RD i+1/2 FRD i 1/2 = 0 i+1/2 =F(u i(x i+1/2 ))+β i+1/2 F RD i 1/2 =F(u i(x i 1/2 )) β i 1/2 i i φ i+1/2 φ i 1/2 MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

159 Relations with other techniques Other examples In (Chou, Shu J.Comput.Phys, 2006 ; Chou, Shu J.Comput.Phys, 2006) the authors propose a WENO Finite Difference scheme consisting of a RD technique using nodal WENO reconstructions instead of Lagrange approximation. The RD formulation permits to keep the simplicity of the WENO FD approach, while allowing high accurate solutions on non-smooth cartesian meshes MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

160 Relations with other techniques Other examples In (Abgrall, Shu Comm.Compu.Phys, 2009) DG schemes are recast as RD. The key is defining the fluctuation including a numerical flux F as φ K = F ˆndl A preliminary construction of a hybrid DG-RD nonlinear scheme is proposed K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

161 Relations with other techniques Other examples Residual distribution schemes based on discontinuous approximation explored in (Hubbard J.Comput.Phys, 2008 ; Abgrall Adv.Appl.Math.Mech, 2010 ; Hubbard, Ricchiuto Computers & Fluids, 2011). As before, the key is defining the fluctuation including a numerical flux F as φ K = F ˆndl RD techniques used to generate nonlinear schemes. The advantages of the discontinuous approximation are retained K MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

162 Relations with other techniques Important points 1. RD as a general framework to study weighted residual discretizations on general meshes 2. RD as a means of constructing non-oscillatory schemes 3. RD to define FV numerical fluxes, interesting applications in presence of source terms (see Part II) MARIO RICCHIUTO - Residual Distribution, Part I (CEMRACS 2012) July 19,

163 .. to be continued... M.R. web : mricchiu BACCHUS Inria team :