Implicit schemes of Lax-Friedrichs type for systems with source terms

Transcription

1 Implicit schemes of Lax-Friedrichs type for systems with source terms A. J. Forestier 1, P. González-Rodelas 2 1 DEN/DSNI/Réacteurs C.E.A. Saclay (France 2 Department of Applied Mathematics, Universidad de Granada (Spain Castro Urdiales, 7-11 Sept., 2009 (Spain

2 Index 1 Introduction 2 Implicit Lax-Friedrichs and local Lax-Friedrichs schemes in the scalar case 3 Implicit Lax-Friedrichs and local Lax-Friedrichs schemes for systems

3 Implicit schemes Are taking more and more importance in the CFD and other areas. Better stability properties. Allows us to employ larger CFL numbers. Reaching the specified instant time with fewer iterations. But with possibly loosing some transient information. So specially appropriate for stationary solutions. Leads to solve large non linear systems of equations. It is convenient to linearize them. Need of efficient linear algebra routines (specially for tri-diagonal and block-diagonal matrices.

4 Index 1 Introduction 2 Implicit Lax-Friedrichs and local Lax-Friedrichs schemes in the scalar case 3 Implicit Lax-Friedrichs and local Lax-Friedrichs schemes for systems

5 Formulation of the Lax-Friedrichs scheme As a variant of the unconditionally unstable FTCS scheme in the approximation of the time derivative, done now as t u ( un 1 +un +1 2 t n, x t = 2un+1 u n 1 un +1 2 t we obtain the so called (explicit Lax-Friedrichs scheme where σ t x = u n n σ ( f f n and independently of n N {0} we have LF LF f f == f +1 f 1 2 u +1 2u + u 1 2σ

6 Lax-Friedrichs numerical flux This results in the following classical L-F numerical flux f LF (u, v = 1 (f (u + f (v 1σ 2 (v u (1 where f 1 2 with u n f LF ( u 1, u, f+ 1 2 f± 1 2 = 1 2 u ( t n, x and f n ( f ±1 + f 1 σ f ( u n f LF ( u, u +1. in such a way that it has good stability properties. ( u±1 u (2

7 Rusanov or Local Lax-Friedrichs flux Is ust a variation of the above L-F one, replacing the constant viscosity coefficient ( 1 σ = x t by a new one, calculated locally for each one of the Riemann problems ζ = max f (q q [u,v] (where q [u, v] indicates that q can be lengthwise the segment [u, v] := {q = t u + (1 t v / t [0, 1]} For example, in the case of a convex or concave flux ζ = max { f (u, f (v }.

8 Formulation of the so-called θ-schemes In a similar way, depending on the specific numerical flux f n+1 taken, writing f := f ( 1 1, un+1, we can consider the 2 ( explicit scheme = u n σ f n f n ( implicit scheme n+1 n+1 + σ f f = u n 2 2 or even taking any convex combination of these two, the θ-schemes with θ [0, 1]: ( n+1 n+1 + θ σ f f = u n n (1 θ σ ( f f n (3

9 Formulation and study of the implicit schemes They would lead in general to systems of nonlinear equations that would have to be solved for each time step. Nevertheless, we can develop the expressions in the implicit parts, if we can easily find ṽ n+1 v n+1 in such a way that ( f ṽ n+1 when f ( ṽ n+1 ( f f v n+1 = +1 un+1 1 ( +1 un+1 1 ( +1 ( f 1 +1 un+1 1 f n+1 +1 f n un+1 1 ; but in any case we could also write ( ( f v n+1 +1 un+1 1 = f n+1 n+1 +1 f 1

10 Implicit θ-schemes In the case of considering periodic boundary conditions, the coefficients matrix appearing in each iteration step will have the following form β n 1 γ n α n 1 α n 2 β n 2 γ n α n 1 β n 1 γ n α n β n γ n 0 0 α n +1 β n +1 γ n α n J 1 β n J 1 γ n J 1 γ n J α n J β n J J 1 J = ( δ n 1 δ n 2 δ n 1 δ n δ n +1 δ n J 1 δ n J

11 Implicit θ-schemes So we have, δ n 0 = αn 0 un 1 + βn 0 un 0 + γn 0 un 1 = α n J un J 1 + βn J un J + γn J un J+1 = δn J whereas if the boundary conditions would be of Dirichlet type, also the first and last equations above would not have more sense, and we would have to restrict ourselves to the indices = 1,..., J 1. But both in the case of periodic or Dirichlet boundary conditions, we can apply the same type of reasoning.

12 FTCS and BTCS θ-schemes In the case of a particular θ-scheme, with the centered flux: + σ ( ( 2 θ f v n+1 +1 un+1 1 = u n (1 θ σ 2 f ( v n ( u+1 1 n un δ n (θ the linear system would be clearly tridiagonal α n 1 +βn +γ n +1 = δn (θ, n = 0, 1,... ; = 0, 1,..., J (4 considering periodic boundary conditions ( 0 = J, 1 and un+1 1, n N {0} with un+1 J 1 J+1 α n α n (θ := σ 2 θ f ( v n+1 β n β n (θ := 1 γ n γ n (θ := σ 2 θ f ( v n+1 σ 2 θ f ( ṽ n+1 σ ( 2 θ f ṽ n+1

13 Inversibility and positive definiteness of the matrices We can apply the following Proposition If the square matrix of the linear system is strictly diagonally dominant ( α n + γ n < β n,, and also it is its transpose matrix ( + <,, and the diagonal elements α n +1 γ n 1 β n are positive (β n > 0, then the original matrix not only is invertible, but also definite positive. obtaining a restriction over the time step of the following type: t < x f

14 Lax-Friedrichs θ-schemes In this case the tridiagonal system reads α n α n (θ := θ ( ( σ f v n θ ( 2 β n β n (θ := 1 + θ γ n γ n (θ := θ ( ( σ f v n θ ( 2 σ f ( ṽ n+1 σ f ( ṽ n We also can see that the system (for θ 0 is not symmetric, since γ n := θ ( ( σ f v n α+1 n := θ ( ( σ f v n γ n α+1 n = σ ( ( ( 2 θ f v n+1 + f v n+1 n+1 +1 = σ θ f ( v +1/2 [ ] for some v n+1 +1/2 v n+1, v n+1 +1 (if we suppose f continuous.

15 Lax-Friedrichs θ-schemes But always that α n and γ n have the same sign: α n, γn 0 (because 0 α n, γn is not possible since it would imply the ( ( following contradiction σ f v n+1 1 < 1 σ f v n+1 ; ( so 1 σ f x 1 (equivalent to t it will v n+1 also be verified that 0 α n + γ n = θ < 1 + θ = β n f ( v n+1 β n whereas in the case that α n and γ n have distinct sign, it would be necessary to study the situation (suposing in any case θ 0, because the case θ = 0 is trivial, since then the scheme is reduces to the explicit case.

16 Lax-Friedrichs θ-schemes Proposition In order that the matrix of coefficients of the linear system that appears in the application of this schemes, would be strictly diagonally dominant it would suffice that: As 1+θ θ or t x f or always that t > t < 1+θ θ f ( x v n+1 ( x f v n+1 for some, then > 1 for every θ ]0, 1], a sufficient condition of invertibility of these matrices would be the following bound t < 1 + θ θ x f (5

17 Lax-Friedrichs θ-schemes Using the Proposition 1, we can also study the diagonal dominance of the corresponding transpose matrix, Case Always that α+1 n and γn 1 will not have the same sign and ( ( ( f v n+1 + f = 2 f will not vanish then we will 1 v n+1 +1 v n+1 have a restriction over the time step than can be used in order to maintain the strict diagonal dominance of the transpose matrix of the implicit scheme of the form t < ( θ f ( x v n+1

18 Lax-Friedrichs θ-schemes Case But if α+1 n and γn 1 may have both the same sign then this restriction will take the following forms ( t < x (, if α n θ f v n+1 +1, γn 1 0 t < 1 θ x f ( v n+1, if α n +1, γn 1 0 It would be also interesting to investigate now in which practical situations of the application of this implicit scheme to a concrete problem we would have to deal with each of these cases, but it would be the subect of other part of the study, when we will apply this scheme to concrete examples.

19 Rusanov or Local Lax-Friedrichs θ-schemes We can rewrite now this implicit scheme σ ( ( 2 θ f v n+1 ( ζ n+1 1/ σ ( 2 θ ζ n+1 + σ ( ( 2 θ f v n+1 +1/2 + ζn+1 1/2 ζ n+1 +1/2 +1 = δ n (θ f ( v n ( u+1 u n (1 θ σ 1 n un ( 2 +ζ 1/2 n u n u 1 n ( u+1 n un ζ n +1/2

20 Rusanov or Local Lax-Friedrichs θ-schemes Or in a tridiagonal way, α n 1 +βn +γ n +1 = δn (θ, n = 0, 1,... ; = 0, 1,..., J identifying appropriately the coefficients α n α n (θ := σ ( ( 2 θ f v n+1 ζ n+1 1/2 σ ( ( 2 θ f ṽ n+1 β n β n (θ := 1 + σ ( 2 θ ζ n+1 +1/2 + ζn+1 1/2 γ n γ n (θ := σ ( ( 2 θ f v n+1 σ ( 2 θ such that, for all and n ζ n+1 +1/2 f ( ṽ n+1 0 < 1 = α n (θ + β n (θ + γ n (θ β n (θ, θ [0, 1] ζ n+1 1/2 ζ n+1 +1/2

21 Rusanov or Local Lax-Friedrichs θ-schemes We obtain easily the relations, for all and n 0 < 1 = α n (θ + β n (θ + γ n (θ β n (θ, θ [0, 1] 0 < β n (θ = β n (θ and α n (θ + γ n (θ 0; in such a way that, if we know the no possitivity of both coefficients, α n, γn 0 0 α n + γ n = α n γ n = σ ( 2 θ < 1 + σ 2 θ ( ζ n+1 +1/2 + ζn+1 1/2 = β n = ζ n+1 1/2 + ζn+1 +1/2 we also could assure the strict diagonally dominance of the matrices involved, so that we also have their invertibility and the good setting out of the implicit scheme and a positivity result (positive definite matrix. β n

22 Rusanov or Local Lax-Friedrichs θ-schemes If we know that α (θ and γ (θ have the same sign, we would have that Case In the case that both are non positive 0 α n (θ + γ n (θ = σ ( 2 θ ζ n+1 < 1 + σ ( 2 θ ζ n+1 1/2 + ζn+1 +1/2 = β = β 1/2 + ζn+1 +1/2 having assured in this way the good setting out of the corresponding implicit scheme (for θ ]0, 1], whereas for θ = 0 we would obtain the identity matrix and we would recover the usual explicit scheme.

23 Rusanov or Local Lax-Friedrichs θ-schemes Case In the case that one of then were non negative (for example α n (θ 0 we can see easily that this would imply the non positiveness of the other because ( α n (θ 0 f v n+1 ( f v n+1 ζ n+1 1/2 0 0 ζ n+1 +1/2 γn (θ 0 (and the same can be easily proved for the implication γ n (θ 0 = α n (θ 0 so that if we assume the non negativeness of both coefficients, this will lead to a contradiction, unless both vanish, that also imply that 1 = un+1 = +1 would be a sonic point of the flux function (where f (q vanish.

24 Rusanov or Local Lax-Friedrichs θ-schemes So, we can also ask to ourselves the following question, In which situations we will have that α n and γ n take values of distinct sign? We have done an exhaustive study of this question, arriving to the following bounds for the time steps to be taken in these cases (having assured previously the not vanishing of the denominator: case 0 < γ n then t < ( θ (f x v n+1 ζ n+1 +1/2 case 0 < α n we must have then t < ( θ ( f x v n+1 ζ n+1 1/2

25 Index 1 Introduction 2 Implicit Lax-Friedrichs and local Lax-Friedrichs schemes in the scalar case 3 Implicit Lax-Friedrichs and local Lax-Friedrichs schemes for systems

26 Implicit Lax-Friedrichs schemes In the case of hyperbolic systems of conservation laws with source term t U + x f (U = g (U, x U R m the expression of the L-F and local L-F (Rusanov schemes would have the same form that in the scalar case and ĝ (U n 1, Un, Un +1 ;σ should relate adequately with the original function g in the source term. U n+1 = U n σ ( fn f n + ĝ (U n , Un, Un +1 ;σ 2 2 using the corresponding numerical fluxes.

27 Implicit Lax-Friedrichs schemes The well-known Lax-Friedrichs numerical flux reads flf (u, v = 1 (f (u + f (v 1σ 2 (v u (6 Whereas the Rusanov s numerical flux will read as follows fru (u, v = 1 (f (u + f (v ζ (v u, (7 2 where ζ := max (ρ (Df (q, for q [u, v], the segment oining both states. ρ (Df (q is the corresponding espectral radius of the Jacobian matrix of f evaluated in q, that we will also notates Df (q A (q.

28 Implicit Lax-Friedrichs schemes The implicit version for systems of the Lax-Friedichs type schemes would write as usual ( U n+1 + σ fn+1 f n+1 = U n ĝ (U n 1, Un, Un +1 ;σ 2 2 (8 or, if we also evaluate the source terms ĝ at time n + 1 and not at time n ( ( U n+1 + σ fn+1 f n+1 + ĝ U n , Un+1, U n+1 +1 ;σ = U n (9 2 2 This last option (9 will leads to the resolution of large systems of non linear equations in each time step, and will not be considered for the moment.

29 Implicit Lax-Friedrichs schemes On the other hand, the first option (8 can be developed using the expression (6 or (7, of the corresponding numerical flux = U n+1 + σ ( f n+1 2 = ( σ ( 2 A taking V n+1 ( f U n fn+1 1 V n+1 U n + ĝ (U n 1, Un, Un +1 ;σ σ ( 2 ζ U n Un+1 + U n+1 1 σ 2 ζ I ( σ ( + 2 A U n+1 1 V n+1 + (1 + σ ζ Un+1 σ 2 ζ I the corresponding Roe s state, satisfying ( ( ( f U n+1 1 = A V n+1 U n+1 +1 Un+1 1 U n+1 +1

30 Implicit Lax-Friedrichs type schemes So, in order to try to show the inversibility of the corresponding matrices associated to the application of these implicit schemes, following the ideas of partitioned matrices explained in the book of Varga: Geršgorin and His Circles (see also the classical articles of Feingold&Varga-1962 and Varah-1972, as partitioned as follows (not considering into account for the moment boundary conditions A A, 1 A, A,

31 Implicit Lax-Friedrichs schemes where, these m m matrix blocks are the following (for adequate subindexes A, := (1 + θ σ ζ I ( A, 1 := σ 2 (A θ ( A,+1 := σ 2 (A θ V n+1 V n+1 ζ I + ζ I so that, to have the strictly block diagonally dominance of the matrix A (according to the definition in Varga it will suffice to take any of the matrix-norms usually taken, that come from the corresponding vector norms for p = 1, 2 or respectively.

32 Partitioned matrices The interesting thing is that this prominent role of the diagonal entries of a matrix can be generalized also for block matrices, through the use of partitions of C n (or R n in the case of real matrices. By a such partition we mean a direct decomposition in pairwise disoint linear subspaces, each having at least dimension one, whose direct sum is still C n (or R n. We will consider a general partition of C n = W 1... W l with p { p } l =0 such that the nonnegative integers (p N satisfy that p 0 := 0 < p 1 < p 2 <... < p l := n assuming that, for all L {1, 2,..., l}, W = span { e k : p k p } (where e k ( δ k,1,..., δ k,n denotes the usual column basis vectors in C n or R n ; and the Kronecker symbol δ k, := 1 only if k = while vanish otherwise.

33 Partitioned matrices So, given any matrix A ( a i, C n n and a partition p { } l p =0 of Cn = W 1... W l, the matrix results partitioned according to p as A 1,1 A 1,2... A 1,l A 2,1 A 2,2... A 2,l A =.. ( A i, A l,1 A l,2... A l,l i, L (10 Each block submatrix A i, C (p i p i 1 (p p 1 still can represents a linear transformation from W to W i for every i, L (with dim W = p p 1 1, L.

34 Partitioned matrices We can also consider a norm l-tuple φ (φ 1,..., φ l, according to the given partition p { } l p =0, where each φ ( L is a norm on the corresponding subspace W. We will denote the collection of all such norm l-tuples associated with p simply by Φ p, so that we will also have the usual matrix norms associated, for any, k L ( A,k φ φ A,k x ( = sup = sup φ A,k x (11 x W k φ k (x x W k x 0 φ k (x=1 Anyway, we always can consider the more used matrix norms, that come from the well-known corresponding vector norms for p = 1, 2 or, respectively.

35 Partitioned matrices For each block diagonal submatrix A i,i (i L we have m ( ( φ i Ai,i x A i,i := inf (12 x W i φ i (x x 0 so that if A i,i is non-singular then m ( A A i,i = 1/ 1 and φ m ( A i,i = 0 otherwise. This quantity (12 can be also called the reciprocal norm of A i,i. We set for any i L (with the convention that r φ (A := 0 if l = 1 r φ i,p (A := L\{i} i,p i,i A i, φ. (13

36 Partitioned matrices We can now give the main generalizations by Feingol&Varga in the case of partitioned matrices Definition Given a partition p of C n and a norm l-tuple φ Φ p then a matrix A = ( A i, partitioned following p will be said strictly i, L block diagonally dominant with respect to φ if Theorem m ( A i,i > r φ i,p (A, i L. (14 Given a matrix A = ( A i, partitioned following the partition i, L p of C n and a norm l-tuple φ Φ p then, if A is strictly block diagonally dominant with respect to φ we can also assure that A is nonsingular.

37 Implicit Lax-Friedrichs schemes In the L-F case (using now that m ( A, = 1 + σ θ ζ = 1 + θ, and, for p { } l p := {0, m,..., m} and choosing the l-tuple =0 norm φ (φ 1,..., φ l, l copies of any of the usual vector and matrix norms (for p = 1, 2 or then = θ σ 2 ( A ( V n+1 r φ,p (A := k L\{} A,k φ ( + ζ I + A V n+1 ζ I p p θ σ ( ( 2 A V n+1 p + 2ζ I 2 p ( ( p = θ 1 + σ A V n+1

38 Implicit local Lax-Friedrichs θ-schemes Now the m m matrix blocks are the following (for adequate subindexes ( A, := (1 + σ 2 θ ζ n+1 +1/2 + ζn+1 1/2 I ( A, 1 := σ 2 (A θ V n+1 + ζ n+1 ( 1/2 I A,+1 := σ 2 (A θ V n+1 ζ n+1 +1/2 I Now m ( σ ( A, = θ ζ n+1 +1/2 + ζn+1 1/2, obtaining ust the same restriction as in the case of the Lax-Friedrichs θ-scheme for systems t < 1 θ max x ( A V n+1 p.

39 Conclusions An exhaustive theoretical study of well-posed implicit θ-schemes of Lax-Friedrichs type. Invertibility and positive definiteness of the matrices in the scalar case. With explicit time step restrictions. That can be also determined for concrete problems. Use of partitioned matrices and generalizations of the Geršgorin circles theorem in the system case. The source terms do not pose special problems a priori. We are doing also many practical calculations to see other subtleties in concrete problems. All this altogether will complete the present theoretical study.

40 Bibliography D. G Feingold, R. S. Varga, Block Diagonally Dominant Matrices and Generalizations of the Gerschgorin Circle Theorem, Pacific Journal of Mathematics, Vol. 12 (1962, pp E. Godlewski, P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences-Springer (1996. R. J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics-Cambridge Univ. Press (2003. R. Varga, Gerschgorin and His Circles, Springer Series in Computational Mathematics, 36 (2004. J. M. Varah, On the Solution of Block-Tridiagonal Systems Arising from Certain Finite-Difference Equations, Mathematics of Computation, Vol. 26 Number 120 (1972, pp

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