Control Errors in CFD!

Transcription

1 Control Errors in CFD! Bernard Müller Department of Scientic Computing, Information Tecnolog, Uppsala Universit, Box 337, S Uppsala, Sweden, bernd@tdb.uu.se, URL ttp:// bernd/ Abstract Error control in computational uid dnamics (CFD) as been crucial for reliabilit and ecienc of numerical ow simulations. Te roles of truncation and rounding errors in difference approximations are discussed. Truncation error control is reviewed for ODEs. For dierence approximations of PDEs, discretization error control b Ricardson extrapolation is outlined. Applications to anisotropic grid adaptation in CFD are sown. Alternative approaces of error control in CFD are mentioned. 1 Introduction Computational uid dnamics (CFD) as been maturing using wind tunnel measurements for validation. However, experiments are often not available, in particular in new multidisciplinar applications. Terefore, te assessment of matematical models as become crucial. Tus, numerical errors in te discretization of matematical models ave to be minimized. Error control is essential for te reliabilit of CFD. Moreover, error control is te prerequisite for adaptive metods, wic are decisive for ecienc. Eriksson et al. [] cite te pilosoper Ludwig Wittgenstein ( ): On wat ou cannot compute wit error control, ou must be silent. Since Wittgenstein was an aeronautical engineer before turning to pilosop, e was well aware of te need for reliable computations in aeronautics. Truncation and Rounding Errors Let us rst consider te approximation of te rst derivative df(x) of a given function f(x) b nite dierences. As an example, we take te second-order central dierence approximation D f(x) = f(x + ) f(x ) (1) were is te step size. Inserting te Talor series expansions around x into (1), we obtain te truncation error T (x) := D f(x) df(x) = f (x) +O( 4 ) 6 () sowing te second-order accurac. Te truncation error indicates ow to get iger accurac: -renement Te grid size is reduced. For example, if te grid size is alved, i.e. is used instead of, te truncation error of te second-order central dierence approximation is reduced b a factor of = 1 4, 1 i.e. T (x) T (x). p-renement Te order of accurac p is increased. E.g. instead of te second-order central dierence approximation wit p =, we can use a iger order metod, e.g. p = 4, for wic te truncation error is lower. For a dierence metod of order p, te truncation error will be reduced b a factor of p, wen te grid size is alved. Tus, te accurac is te more increased b grid renement te larger p is. However, no matter wat te order of p is, we cannot see te expected error reduction as. Ten, we even observe an increase of te error. Te reason for te error increase as is caused b rounding errors, wic lead to cancellation errors. Te errors of second-, fourt- and sixt-order central difference approximations of dexp() are sown in 1

2 1 Error of nd, 4t and 6t order central FDM for d exp() / d x 1 nd order FDM 4t order FDM 6t order FDM 1 Inlupp 1,. central single central double error log(absolute error of FDM for d exp() / d x log() Figure 1: Error of second-, fourt- and sixt-order central dierence approximations in double precision. of dexp() Figure : Error of second-order central dierence approximation of dexp() wit single and double precision. Fig. 1. Te nite dierences were computed in double precision. Te error of D exp(), te second-order central dierence approximation of dexp(), is sown in Fig.. Te inuence of single and double precision on te rounding error is clearl visible. An real number u is represented as a oating point number fl(u) on te computer. Te following relation olds: were ɛ ɛ m fl(u) = u(1 + ɛ) { 1 7 for single precision 1 16 for double precision were ɛ m is macine epsilon. Wen computing te second-order central dierence approximation, te dominant rounding error is te cancellation error caused b computing te dierence of almost equal numbers. Neglecting te error caused b te oating point division b, we get on te computer instead of (1) D fl(f(x)) = = (D f(x))[1 + were f = f(x + ) f(x ) and ɛ ± te relative error of fl(f(x ± )). We see tat te relative cancellation error is large, if f(x±) f are large and cannot be reduced sucientl b multiplication b ɛ ±, for wic ɛ ± ɛ m olds. Tus, cancellation errors and rounding errors in general can be better kept under control b using double precision tan b single precision. However, even wit double precision, we sould be aware of rounding errors, wic can pollute numerical results, cf. Fig.s 1 and. In CFD, cancellation errors occur, wen we are dealing wit ows, for wic te canges are minute compared wit te mean ow. An example is te computation of compressible low Mac number ow. Wen solving te compressible Euler and Navier-Stokes equations for Mac numbers M 1, te termodnamic quantities like densit ρ, pressure p and temperature T onl cange ver sligtl compared wit teir stagnation values. Te consequence is tat te computation of e.g. p can lead to large cancellation errors, wic make te results useless. Fig. 3 fl(f(x + )) fl(f(x )) sows te expansion wave in a tube computed f(x + ) ɛ + f(x ) wit te axismmetric Navier-Stokes equations at low Mac numbers [1], [9]. Te ɛ ] f f re-

3 (x,) extrapolated value (x,) Figure 3: Erroneous pressure distribution of an expansion fan in a tube computed wit conventional formulation of te Navier-Stokes equations leading to cancellation errors in single precision at M = O(1 6 ) [1], [9], : computed result, : analtical solution. sults wit single precision are completel erroneous owing to cancellation errors. However, if te Navier-Stokes equations are formulated in terms of te perturbation variables wit respect to stagnation variables like ρ = ρ ρ, te cancellation errors can be minimized [1], [9]. Computations at M = O(1 1 ) can be performed wit single precision using te perturbation formulation, wic is matematicall equivalent to te original Navier-Stokes equations but formulated in a form better suited for computation, cf. [1], [9]. You simpl solve for e.g. densit cange instead of densit wit te same conservation laws. 3 Truncation Error Control for ODEs Let us consider an initial value ordinar differential equation (ODE) { = f(x, ), x > () = (3) were x denotes time and = (x) is te unknown function to be determined. We want to control te step size of a numerical metod to solve ODE (3) based on te local truncation Figure 4: Basic idea of Ricardson extrapolation. error. We sall consider Ricardson extrapolation and Runge-Kutta-Felberg metods. 3.1 Ricardson Extrapolation Te Ricardson extrapolation is a general metod to improve te accurac of numerical results. Te basic idea of extrapolating results wit step sizes and to = can be summarized as follows, cf. Fig. 4: A metod of order p is used, meaning tat we know te teoretical beaviour of te truncation error as. Compute a solution in x k using step size : (x k, ) Compute a solution in x k using step size : (x k, ) Given two values wit dierent error and knowing tat te truncation error beaves as O( p ) it is possible to extrapolate an approximation of te exact value (x k, ). For a metod of order p, te exact value (solution) can be expressed as (x k ) = (x k, ) + c p + O( q ), 3

4 were c is a constant and p and q are integers wit p < q. Solving te problem wit step sizes and ields (x k ) = (x k, ) + c p + O( q ) (x k ) = (x k, ) + c() p + O( q ) Subtracting tese equations, we obtain te leading term of te truncation error c p = (x k, ) (x k, ) p 1 + O( q ). (4) Tus, adding te leading error term to (x k, ), we obtain a better approximation of te exact solution (x k ), because (x k ) = (x k, )+ (x k, ) (x k, ) p +O( q ) 1 Te algoritm for truncation error control based on Ricardson extrapolation becomes: 1. Compute k using a metod of order p wit step sizes and.. Estimate te truncation error b te leading term c p wit Ricardson extrapolation (4). 3. If error c p < tol were tol is a prescribed tolerance, accept k. Compute new step size k+1, were k+1 > k. 4. If error c p > tol, discard k and compute new k less ten te previous one. Compute k. Continue until error c p < tol. If te truncation error is T (x k ) c p, we coose te new step size k+1 suc tat te estimated truncation error is approximatel equal to te desired tolerance tol, i.e. T (x k+1 ) tol. Tus, te new step size is computed from k+1 = ( ) Θ tol 1/p k, T (x k ) were Θ < 1 is a safet factor. 3. Runge-Kutta-Felberg Metods Instead of computing (x k, ) wit two dierent step sizes, one can compute (x k, ) wit two metods aving dierent orders of accurac. Te dierence of te two results can be used to estimate te truncation error. Te computational work is not doubled, if Runge- Kutta-Felberg metods are used. For example, 7 stages are performed and two dierent weigtings of te stages ield two metods of orders 4 and 5, respectivel. Ten, te truncation error can be estimated b te dierence of te two results. Tis error control is e.g. used in te MATLAB ODE-routine ode45. 4 Discretization Error Control for PDEs Consider an initial boundar value problem for a partial dierential equation (PDE) like te Navier-Stokes equations Lu =, (5) were L denotes te dierential operator and u = u(x, t) is te unknown function to be determined. Here, x and t denote te spatial coordinate and time, respectivel. As an example, we consider te scalar linear 1D advection equation wit initial and boundar conditions u t + au x =, x [, 1], t >, (6) u(x, ) = f(x), x [, 1], u(, t) = g(t), t, were a > is a constant advection velocit. We coose a nite dierence metod to discretize te PDE (5) L u =, (7) e.g. te rst-order explicit upwind sceme wit v = u for (6) v n+1 j v n j t + a vn j vn j 1 x =, (8) 4

5 were j = 1,..., N = 1 x, n =,..., t t 1. Insert te exact solution u of te PDE (5) into te FDM (7) to obtain te truncation error T = L u = L u Lu. (9) Inserting u(x j, t n ) for v n j into (8), we obtain T (x j, t n ) = u tt(x j, t n ) t + O( t ) a u xx(x j, t n ) x + O( x )(1) = c x + O( x ) + d t + O( t ) Te task is to estimate te leading error terms, i.e. in our example c x and d t. Following Ferm and Lötstedt [3] and Lötstedt et al. [8], we can use Ricardson extrapolation to approximate te truncation error b te discretization error: 1. Compute v n+1 b solving L v n = on ne grid wit step size = x and time step k = t.. Restrict te ne grid solution v to te coarse grid wit step size, i.e. use restriction operator (N assumed even) I vn = [vn, vn, vn 4,..., vn N ]T. Compute discretization error τ = L I vn (11) on te coarse grid wit time step k, i.e. in te example τ (x j, t n ) = vn+1 j vj n t j =, 4,..., N. + a vn j vn j, x (1) 3. Approximate te dierence of te truncation errors on te ne and coarse grids b te dierence of te discretization errors on te ne and coarse grids (same k = t). Ten, te leading spatial truncation error is approximated b Ricardson extrapolation: c p τ = L I vn I L v n p 1. (13) 5 In our example, we ave p = 1. Since L v n =, te rigt and side of (13) is te dierence of te spatial discretizations on te coarse and ne grids, i.e. for our example c τ (x j, t n ) = a vn j vn j a vn j vn j 1 x x 4. Ceck discretization error τ and rene or coarsen grid, if needed. Recalculate starting from 1. until te tolerances tol are met. Tus: Wile τ tol refine, rene grid, i.e. use. Wile τ tol coarsen, coarsen grid, i.e. use. 5. After completing te grid adaptation in computing v n+1 wit te required level of te spatial discretization error, control temporal truncation error as for ODEs, cf. section 3. 5 Anisotropic Grid Adaptation In multi dimensions, te discretization errors in eac coordinate direction are computed b 1. coarsening in x l -direction, l = 1,, 3 in 3D,. computing te dierence approximations of te x l -derivatives, l = 1,, 3, on coarse grids using te ne grid solution, 3. comparing wit ne grid discretizations of te respective derivatives. If te error tolerance for renement (coarsening) is not met, ceck weter renement (coarsening) in one, two or all tree x l - directions is needed. Ten, an anisotropic grid is obtained. Te procedure can be easil extended to - nite dierence metods on general, i.e. non- Cartesian, structured grids. Te extension to structured nite volume metods is outlined.

6 1.5 u u x. Figure 5: Anisotropic adapted grid wit u velocit contours in te transformed domain, adaptation based on discretization error control for incompressible ow over a at plate at Re = 5 b Ferm and Lötstedt [3]. in detail b Ferm and Lötstedt [3]. Dening coarse grid cells b te union of ne grid cells allows to generalize Ferm and Lötstedt's approac to unstructured nite volume metods. Te data structure is considerabl simplied, if te renement and coarsening is not done locall, but blockwise. Terefore, Ferm and Lötstedt use anisotropic grid adaptation wit a block-structured nite volume metod for te D compressible and incompressible Navier-Stokes equations [3]. Examples b Ferm and Lötstedt [3] are presented ere. Stead incompressible ow over a at plate at Renolds number Re = 5 is computed using te articial compressibilit metod. Wile te initial grid ad 34 cells, te adapted grid wit cells is sown in Fig. 5 togeter wit te u- velocit contours. Te renement tolerance was tol refine =.17. Note tat te - coordinate is stretced in Fig. 5. Te u- velocities and te corresponding errors are compared at dierent stations in Fig.s 6 and 7, respectivel. Compared to te initial grid, te error is considerabl reduced wit te adapted anisotropic grid Figure 6: u velocit proles (smbols ) in transformed -coordinate compared wit te Blasius solution (solid lines), adaptation based on discretization error control for incompressible ow over a at plate at Re = 5 b Ferm and Lötstedt [3]. u Figure 7: Error of u velocit contours in te transformed domain wit initial grid (solid lines) and adapted grid (dotted lines), adaptation based on discretization error control for incompressible ow over a at plate at Re = 5 b Ferm and Lötstedt [3]. 6

7 outflow, < outflow, > 1 TE LE TE Figure 8: Anisotropic adapted grid wit u velocit contours in te transformed domain, adaptation based on discretization error control for laminar subsonic ow over a NACA1 airfoil b Ferm and Lötstedt [3] LE and TE denote leading and trailing edges, respectivel x Fig. 8 sows te adapted grid based on discretization error control for laminar subsonic ow over a NACA1 airfoil at 1.5 angle of attack, Mac number M =.5 and Renolds number Re = 3 b Ferm and Lötstedt [3]. Te contour plots in Fig.s 9 and 1 were obtained b tat approac wit te renement tolerance tol refine =.58, 16 cells in te initial grid and 96 cells in te adapted grid (Fig. 8). Te analsis of te dependence of te solution error u u on te discretization error τ and te results b Ferm and Lötstedt [3] indicate tat te discretization error is a useful error indicator. 6 Conclusions A subtle analsis of certain ows, e.g. boundar laer ow, reveals te relation between te discretization errors and te errors of te ow variables [3]. Te conclusion is tat it often suces to control te discretization errors. Oter a posteriori error estimates are based on Talor expansion [4], [5], [7]. u(x) = u()+x T u()+ 1 xt ( T u())x+... (14) Figure 9: u velocit contours wit anisotropic grid adaptation based on discretization error control for laminar subsonic ow over a NACA1 airfoil b Ferm and Lötstedt [3] x Figure 1: Pressure contours wit anisotropic grid adaptation based on discretization error control for laminar subsonic ow over a NACA1 airfoil b Ferm and Lötstedt [3]. 7

8 Te adjoint metod optimizes te grid for solving a PDE b solving te adjoint problem. Tis approac is used wit te Galerkin nite element metod in [6], [], [11], [1]. Oter errors like oscillations at moving uid interfaces in gas mixtures [9], ux discretization errors due to source terms [9], numerical instabilities due to te 'carbuncle' penomenon [1], etc. are fundamental to te numerical metods used and require numerical analsis to understand te reasons for te errors and devise cures. Te most important recommendation is to be critical to te results produced b a CFD code and to use not onl psical insigt but also all te available knowledge on numerical analsis to get reliable results ecientl. Terefore, control errors in CFD! ACKNOWLEDGEMENTS Te contributions b m colleagues Lars Ferm and Per Lötstedt are gratefull acknowledged. I tank Hideaki Aiso, NAL, Japan, for te invitation to te t ANSS and Yoko Takakura for te ospitalit at Toko Noko Universit. References [1] Becker, R., Rannacer, R.: An Optimal Control Approac to a posteriori Error Estimation in Finite Element Metods. Preprint 1-3, IWR Heidelberg, 1. [] Eriksson, K., Estep, D., Hansbo, P., Jonson, C.: Introduction to Adaptive Metods for Dierential Equations. Acta Numerica (1995), pp [3] Ferm, L., Lötstedt, P.: Anisotropic Grid Adaptation for Navier-Stokes' Equations. Tecnical Report -3, Department of Information Tecnolog, Uppsala Universit, June. [4] Jasak, H., Gosman, A.D.: Automatic Resolution Control for te Finite Volume Metod. Part 1: A-posteriori Error Estimates. Submitted to Numerical Heat Transfer,. [5] Jasak, H., Gosman, A.D.: Automatic Resolution Control for te Finite Volume Metod. Part : Adaptive Mes Renement and Coarsening. Submitted to Numerical Heat Transfer,. [6] Jonson, C.: On Computabilit and Error Control in CFD. International J. for Numerical Metods in Fluids, Vol. (1995), pp [7] Lepage, C.Y., Sueric-Gulick, F., Habasi, W.G.: Anisotropic 3-D Mes Adaptation on Unstructured Hbrid Meses. AIAA Paper -859, 4t ASM&E, Jan., Reno, Nevada. [8] Lötstedt, P., Söderberg, S., Ramage, A., Hemmingsson-Fränden,L.: Implicit Solution of Hperbolic Equations wit Space- Time Adaptivit. BIT, Vol. 4 (), pp [9] Müller, B., Jenn, P.: Improving te Reliabilit of Low Mac Number Flow Computations. CFD J., Vol. 9, No. 1 (1), pp [1] Pandol, M., D'Ambrosio, D.: Numerical Instabilities in Upwind Metods: Analsis and Cures for te Carbuncle Penomenon. J. Comput. Psics, Vol. 166 (1), pp [11] Rannacer, R.: Error Control in Finite Element Computations. In Bulgak, H. and Zenger, C. (eds), Proc. Summer Scool Error Control and Adaptivit in Scientic Computing, Kluwer Academic Publisers (1998), pp [1] Sesterenn, J., Müller, B., Tomann, H.: On te Cancellation Problem in Calculating Compressible Low Mac Number Flows. J. Comput. Psics, Vol. 151 (1999), pp