A Demonstration of the Advantage of Asymptotic Preserving Schemes over Standard Finite Volume Schemes

Transcription

1 A Demonstration of te Advantage of Asymptotic Preserving Scemes over Standard Finite Volume Scemes Jocen Scütz Berict Nr. 366 Juni 213 Key words: conservation laws, asymptotic metods, finite volume metods, finite element metods. AMS Subject Classifications: 35L65, 76M45, 65M8, 65M6 Institut für Geometrie und Praktisce Matematik RWTH Aacen Templergraben 55, D 5256 Aacen (Germany

2 A DEMONSTRATION OF THE ADVANTAGE OF ASYMPTOTIC PRESERVING SCHEMES OVER STANDARD FINITE VOLUME SCHEMES. JOCHEN SCHÜTZ June 7, 213 Abstract. We apply te concept of Asymptotic Preserving (AP scemes [17] to te linearized p system and discretize te resulting elliptic equation using standard continuous Finite Elements instead of Finite Differences. Te fully discrete metod is analyzed wit respect to consistency, and we compare it numerically wit more traditional metods suc as Implicit Euler s metod. Numerical results indicate tat te AP metod is indeed superior to more traditional metods. 1. Introduction. We consider te p system [12] wit a linear pressure function p(v := 1 2 v and a rigt-and side g, v t u x = (x, t Ω R + (1.1 u t + p(v x = g(x, t (x, t Ω R + (1.2 on a domain Ω R subject to suitable initial and boundary values, were for simplicity we coose te latter to be v(x = x Ω. In a (simplified pysical application, u and v could denote velocity and (variations of te specific volume of te fluid. Obviously, te equation can be written as w t + f(w x = G(x, t (x, t Ω R + (1.3 for w := (v, u T, f(w := ( u, 1 v T and G(x, t := (, g(x, t T. 2 Te eigenvalues of te Jacobian of te flux function f are ± 1, and so a fully explicit Finite-Volume sceme will not be feasible for small values of, as te timestep will decrease wit. Inspired by Asymptotic Preserving Scemes (AP, we develop a new solver for (1.1-(1.2 based on a combination of Finite Volumes and Finite Elements. Its (fully discrete consistency is investigated, and it is compared wit more traditional numerical scemes wit respect to error versus mes size. We put tis in te simple framework of te p system because it was on a similar system tat Jin [17] derived is famous asymptotic preserving scemes for te first time, and because it is simple (but not too simple, so tat eac step can be easily computed, wic is not te case for more involved systems suc as Euler s equations. As already mentioned, te concept of asymptotic preserving scemes tat we pursue in tis publication as been introduced by Jin [17], building on joint work wit Paresci and Toscani [18]. In tese publications, a sceme is called asymptotic preserving if it is for a consistent sceme for te multiscale limiting equations of (1.1-(1.2 and is stable wit a cfl-number independent of. Tis class of scemes as since been extended to various kinds of equations, suc as, e.g., Euler s equation [13, 3], Sallow-water equations [2] and many more. 1

3 Te current paper is a first attempt to extend te scemes, wic ave been presented for Finite-Volume discretizations, to Galerkin-type scemes. Based on a flux-splitting, we derive an elliptic equation wose diffusion coefficient is dependent on and. Tis equation is solved by continuous Finite-Element metods, and not, as usual, by finite-difference scemes. Te approac, toug it can of course also be written in terms of finite differences, as te advantage tat we can investigate te elliptic equation and its discretization in a rigorous setting in te context of Sobolev spaces. In a first step, we sow tat te elliptic equation is well-posed and uniformly well-conditioned for all values of and. Tis is acieved by introducing problemdependent spaces and norms. In a second step, we restrict ourselves to small and large, i.e., < < 1 and, as it is only in tis setting tat we can use standard Finite-Element scemes [6, 15] instead of stabilized ones [5]. Also for tis setting, we can derive rigorous and uniform (in stability and consistency bounds. Solutions to (1.1-(1.2 tat allow for a limit solution as ave a certain structure (see (2.6-(2.7 in Sec. 2. Our consistency analysis for te fully discrete algoritm eavily relies on tis structure, and we believe tat it is only in tis setting tat one can derive suitable bounds on te consistency error tat do not beave like O( 1 or even worse. As an easy consequence, we can indeed sow tat te proposed sceme is AP. Tis is different to oter autors [13, 2] wo sow tat teir sceme is asymptotic preserving by a Taylor series argument on te semi-discrete stage. Having presented our sceme, we compare it numerically wit two oter scemes. Te surprising outcome is tat te sceme to be presented performs better by orders of magnitude in comparison to more traditional scemes. Te outline of te paper is as follows: In Sec. 2.1, we derive te multiscale limit solution of te linearized p system for. In Sec. 2.2, we split te conservative flux f into a stiff f and a non-stiff f. Based on tis splitting, we derive a semidiscretization in Sec Tis yields an elliptic equation, wic is investigated in Sec Finally, in Sec. 2.5, we formulate te fully discrete algoritm and investigate its consistency in Sec In Sec. 3, we sow numerical results. Sec. 4 offers conclusions and outlook. 2. Asymptotic Preserving Discretization Multiscale limit of te equation. In tis section, we follow a multiscale approac to obtain te limiting equations of (1.3. To tis end, we assume tat our unknown solution (v, u admits a two-scale expansion as v(x, t = v ( (x, t + v (1 (x, t + 2 v (2 (x, t + O( 3 (2.1 u(x, t = u ( (x, t + u (1 (x, t + 2 u (2 (x, t + O( 3. (2.2 Note tat tis approac does not include fast waves, i.e., contributions depending on 1, so one as a uniform limit as. Plugging (2.1-(2.2 into (1.1-(1.2 and balancing te powers of yields tat bot v ( (x, t and v (1 (x, t are independent of x. Terefore, v (1 (x, t can be absorbed into v ( (x, t, and (2.1 reduces to v(x, t = v ( (t + 2 v (2 (x, t. (2.3 Te remaining limiting equations can be easily seen to be v ( t u ( x = (x, t Ω R + (2.4 u ( t v (2 x = g(x, t (x, t Ω R +. (2.5 2

4 A suitable algoritm approximating (1.1-(1.2 for small values of sould, in te vanising limit, be a consistent approximation to (2.4-(2.5. In reference [17], suc a consistency requirement is called asymptotic preserving. For a general conservation law, it is nontrivial to obtain more precise results concerning v ( and u (, see, e.g., [19] for results in te context of Euler s equations. However, in te very simple setting of te linearized p system, we can clarify even more te relation between v and u. Due to te boundary conditions, it can be easily sown tat te only admissible functions are v(x, t = 2 (tξ(x + η(x + O( 3 (2.6 x u(x, t = u ( (t + u (1 + 2 ξ(y dy + O( 3 (2.7 wit a suitable definition of g(x, t := u t (x, t v x (x, t and arbitrary functions ξ, η, u ( and constants u (1, d R. Especially, bot v(x, t and u x (x, t are of order O( 2, wic will elp performing a consistency analysis Flux Splitting. Te way of splitting te flux into stiff and non-stiff parts as an influence on te final algoritm. We coose our splitting according to te following definition: Definition 2.1. Let te flux function f be split into f(w = f(w + f(w. We consider suc a splitting to be admissible if for all < < 1 bot f(w and f(w induce a yperbolic system, i.e., te eigenvalues of bot f (w and f (w are distinct and real, te eigenvalues of f (w are of order one, f(w approaces f(w as 1, and f(w ( approaces f(w for in te sense tat lim 2 f(w f(w =. f(w is called te non-stiff, and f(w te stiff part of te flux function for obvious reasons. To identify stiff and non-stiff parts of te flux function, we make te following ansatz: ( f(w = f(w + f(w α(u =: β( v 2 d ( (1 α(u + 1 β( v 2 Bot α( and β( are yet unknown. One reasonable requirement is α(1 = β(1 = 1, and α( = β( =, so tat one as no stiff contribution given tat is one, and no non-stiff contribution given tat vanises. We make te simple ansatz of α( = a, β( = b. An easy computation sows tat for a, b >, a + b = 2, te eigenvalues of f (w are independent of. A particularly simple coice is a = b = 1, wic we will use trougout tis work. In summary, for tis coice of a and b, we ave f(w = ( u 1 v wit corresponding eigenvalues of te Jacobians ( (1 u, f(w= 1 v 2. λ = ±1, 1 λ = ±. 3

5 2.3. Semi-Discretization. We start te description of our algoritm wit a discretization in time only. For simplicity, we assume tat we work on space-time slabs of (uniform size, altoug uniformity is not a necessary condition. Trougout tis work, we will use standard notation and set w n := w(t n, were t n := n. Based on te flux splitting defined in Sec. 2.2, we obtain a first-order implicit / explicit semidiscretization of (1.3 in time, given by or, in terms of (v, u, w n+1 w n + f(w n x + f(w n+1 x = G n. (2.8 v n+1 v n u n+1 u n = u n x + (1 u n+1 x (2.9 = 1 vn x v n+1 x + g n. (2.1 One way of dealing wit suc a system of implicit equations tat as become a standard ingredient in asymptotic preserving scemes, is to equivalently reformulate (2.9-(2.1 in suc a way tat one obtains an equation for eiter v n+1 or u n+1 alone. We ave decided to formulate an equation for v n+1. To tis end, we note tat (2.1 is equivalent to and plug tis into (2.9: ( 1 u n+1 = u n + vn x vx n+1 + g n, (2.11 ( ( ( 1 v n+1 = v n + u n x + (1 u n + vn x v n+1 ( 1 = v n + u n x + 2 (1 vn xx vxx n+1 + gx n x + g n x (2.12 (2.13 = v n + u n x + 2 (1 vxx n + 2 (1 2 2 vxx n (1 gx n. (2.14 Rearranging terms yields an elliptic equation for v n+1 : 2 (1 2 2 vxx n+1 + v n+1 = v n + u n x + 2 (1 vxx n + 2 (1 gx n. (2.15 Remark 2.2. Using standard teory of partial differential equations, it is easy to see tat (2.15 is a well-posed equation for > and < < 1. However, in standard Sobolev norms, one can see tat only for and 1, one obtains uniform bounds on te condition number of te problem. We will sow in Sec. 2.4 tat one can put te equation in a framework tat allows for a uniform bound on te condition number of (2.15, owever, in a norm tat depends on te diffusion coefficient λ := 2 (1 2. (Wic is, of course, not surprising. 2 Te weak formulation of (2.15 can be cast in a variational framework as a(v n+1, ϕ = ι(ϕ ϕ H 1 (Ω, (2.16 4

6 were a(v n+1, ϕ := ι(ϕ := Ω Ω ( 2 (1 2 2 vx n+1 ϕ x + v n+1 ϕ (v n + u n x ϕ 2 (1 dx and (2.17 ϕ x dx. (2.18 ( v n x + gn Boundedness and coercivity properties of a(, will be discussed in te next sections. Wat concerns ι, we can state te following lemma: Lemma 2.3. Let us assume tat u n u(t n and v n v(t n are functions in H 1 (Ω; g n g(t n is a function in L 2 (Ω; and < < 1. Ten ι H( 1 (Ω. Proof. It is enoug to sow tat bot v n + u n x and 2 v n x (1 + gn are functions in L 2 (Ω, wic is correct because of te assumptions on u n, v n and g n A note on te elliptic equation. Let us now turn to te operator equation (2.16. To make it a well-defined and a uniformly well-conditioned problem for all < 1, we put it in a variational framework wit weigted Sobolev spaces as follows: Definition 2.4. Let te coefficient of te viscous term of (2.16 be denoted by λ, i.e., We define a weigted norm λ as and a corresponding Sobolev-space λ := 2 ( (2.19 ϕ 2 λ := ϕ 2 L 2 + λ ϕ x 2 L 2 (2.2 V λ (Ω := C (Ω λ. (2.21 Corollary 2.5. For λ >, i.e., < 1, te weigted norm λ is equivalent to te standard Sobolev norm, as can be seen form a Poincaré-Friedrics inequality. However, te equivalence constants gets worse as approaces one. Wit tis equivalence in mind, it is easy to see tat { H 1 (Ω, < 1 V λ (Ω = L 2 (Ω, = 1, (2.22 as λ = for = 1 and λ > for < < 1. Remark 2.6. Te weigted norm λ is te energy norm associated to (2.16, i.e., ϕ 2 λ = a(ϕ, ϕ. (2.23 Furtermore, for λ =, te problem (2.16 is not well-posed in H 1 (Ω any more, so te coice of V λ (Ω is actually very natural. Te following lemma computes bot coercivity and boundedness constants of a(, on V λ (Ω: 5

7 Lemma 2.7. Te bilinear form a(, as defined in (2.17 is coercive on V λ (Ω V λ (Ω wit ellipticity constant one, and bounded on V λ (Ω V λ (Ω wit boundedness constant also one. Proof. It is easy to see tat a(ϕ, ϕ = λ ϕ x 2 L 2 + ϕ 2 L 2 = ϕ 2 λ, (2.24 so te bilinear form is elliptic wit ellipticity constant one. Furtermore, using Caucy-Scwartz inequality (tis is possible because of (2.23, one as a(ϕ, ψ ϕ λ ψ λ. (2.25 Te following teorem guarantees tat (2.16 is, for te full range of < 1, a well-conditioned problem: Teorem 2.8. Te equation (2.16 is well-conditioned in V λ (Ω independently of, i.e., for two functionals ι, ι V λ (Ω, and teir corresponding solutions v and ṽ, one as te relation v ṽ λ ι ι V λ. (2.26 v λ ι V λ Proof. It is a classical result from te teory of elliptic pde tat te quotient of boundedness constant and ellipticity constant is indeed te condition number wit respect to a perturbation of te functional ι. Neverteless, for convenience, we give a sketc of te proof. From ellipticity, we can conclude v ṽ 2 λ = a(v ṽ, v ṽ = ι(v ṽ ι(v ṽ ι ι V λ v ṽ λ (2.27 and from boundedness ι V λ = sup ι(u = sup a(v, u v λ. (2.28 u V λ (Ω, u λ =1 u V λ (Ω, u λ =1 (2.27-(2.28 yields ( Full discretization. In tis section, we introduce a fully discrete metod. To tis end, we assume tat our spatial domain Ω is subdivided into cells Ω k as Ω = N x k=1 N x Ω k := [x k, x k+1 ] (2.29 wit midpoints x k. For te ease of presentation (but witout loss of generality, we consider a uniform discretization, i.e., x := x k+1 x k is constant. Using standard conventions, we define wi n to be (an approximation to w(x i, t n. By now, we ave all te ingredients of formulating a fully discrete algoritm for te approximation of (1.1-(1.2 for eac timestep. Te algoritm relies on te following steps: Compute an approximate solution to (2.16 wit (linear Finite-Elements. Update u according to ( k=1

8 Let us make tis more precise: Te rigt-and side ι of (2.16 includes derivatives of bot u and v at time t n. Tese derivatives are approximated using a standard numerical flux function f num for te non-stiff part f of te flux f, in te sense tat ( (u n i x, 1 (vn i x T 1 ( fnum (wi n, w n x i+1 f num (wi 1, n wi n (2.3 As we ave to plug functions in Ω into ι, we assume tat te derivatives are piecewise constant on eac cell. (A better reconstruction is left for future publication, see also Rem Plugging tese quantities into ι yields an approximation ι. Using tis approximation ι, one computes a Finite-Element solution to te elliptic equation, i.e., one computes a solution v n+1 V to were a(v n+1, ϕ = ι (ϕ ϕ V, (2.31 V := {ϕ C (Ω ϕ Ωk is linear for all k; ϕ ( = ϕ (1 = }. (2.32 Subsequently, u n+1 is updated using formula (2.11, employing te approximations of bot (u n i x and (vi n x. Remark 2.9. In tis work, we use a first-order Finite-Volume sceme wit a local Lax-Friedrics flux, given by f num f LF wit f LF (u l, u r := 1 2 (f(u l + f(u r α 2 (u r u l. (2.33 As usual, α denotes te spectral radius of te local Jacobian (wic is, in tis simple setting, a constant. We ave decided to put our investigations in tis simple framework to allow for a fair comparison, wic is not influenced by te coice of limiter or time-integration sceme. Remark 2.1. Note tat for = 1, te elliptic equation (2.16 becomes a triviality and does not ave to be solved. In tis case, te sceme reduces to a standard Finite-Volume sceme wit numerical flux function f num. Terefore, we assume < 1 in te following (Order of Consistency and some stability considerations. In tis section, we sow tat our metod is consistent, and we determine its order of consistency. We ave decided to put tis investigation into te more classical framework of standard H 1 spaces and norms (instead of using V λ, because in tis setting we can use classical Finite-Element spaces and do not ave to rely on stabilized Finite-Elements suc as SUPG. Tis, owever, comes at te price of restricting to < < 1 and. Neverteless, as we are interested in te limit for a moderate time-step, tis is not a severe restriction. To prove consistency of our sceme, we ave to bound te following error parts: e 1 := v(t n+1 v n+1 L 2 (2.34 e 2 := v n+1 v n+1 L 2 (2.35 e 3 := v n+1 v n+1 L 2. (2.36 Te overall consistency error in v, e := v(t n+1 v n+1 L 2, can ten be bounded by te sum of te e i. Let us remind te reader of te following definitions: 7

9 v(t n+1 denotes te exact solution v to (1.3 at time t n+1. v n+1 denotes te exact solution to te elliptic equation, see (2.16. v n+1 denotes te solution to te elliptic equation (2.16 wit rigt-and side ι instead of ι, see Sec v n+1 denotes te Finite-Element solution to te elliptic equation, see (2.31. Let us make te following important assumption wic is motivated by our investigations concerning te multiscale expansion, see (2.6-(2.7: Assumption We assume tat v is given by and tat te spatial derivative of u is given by v(x, t = 2 v (2 (x, t + O( 3. (2.37 u x (x, t = 2 u (2 x (x, t + O( 3. (2.38 Remark Witout tis assumption, it will not be possible to perform a consistency analysis for te small limit, because tere is no limit function as. Tis is very similar to te observation in [19] tat te initial data as to be divergence free to allow for an incompressible limit. We start by bounding e 1. Lemma Te temporal discretization yields te following asymptotic error: e 1 := v(t n+1 v n+1 L 2 = O( 2 2. Proof. By cecking te order of consistency of (2.9, one obtains (t n ξ 1, ξ 2 t n+1 : 1 ( v(t n+1 v(t n u x (t n (1 u x (t n+1 = 1 ( v t (t n v tt(ξ 1 u x (t n (1 (u x (t n + u xt (ξ 2 =v t (t n + 2 v tt(ξ 1 u x (t n (1 u xt (ξ 2 (1.1 = 2 v tt(ξ 1 (1 u xt (ξ 2 Ass.2.11 = O( 2, wic yields indeed te desired order of accuracy. Let us continue by bounding e 2. w n denotes te (assumed smoot exact solution w = (v, u T at time t n. By w n x, we denote te exact derivative of w at time t n, and by w n x, we denote te approximation of te derivative by numerical flux functions. It is known tat w n x w n x L 2 = O( 2 x, (2.39 wic, again, is a consequence of Ass Before considering te full approximation error, we ave to turn to te operator equation (2.16 again in te context of classical Sobolev-spaces. Following standard convention, we define te H 1 norm to be ϕ H 1 := ϕ x L 2, (2.4 8

10 and remind te reader of Poincaré-Friedric s inequality ϕ L 2 C P F ϕ H 1. (2.41 We start wit te following teorem tat guarantees tat (2.16 is, also for small, easy to solve. Teorem For a given < 1, let <, and. Te equation (2.16 is well-conditioned in H 1 independently of, i.e., for two functionals ι, ι H 1 (Ω, and te corresponding solutions v and ṽ, one as te relation v ṽ H 1 v H 1 M γ ι ι H 1, (2.42 ι H 1 and M γ can be bounded uniformly in terms of. Proof. It is easy to see tat a(, fulfills, for < 1, an ellipticity condition on H 1 wit ellipticity-constant γ, and it is a bounded bilinear form wit stability constant M. Bot γ and M can be explicitly given as γ = 2 (1 2 2, M = γ + C 2 P F. (2.43 Te rest of te proof goes along te lines of Tm Note tat te quotient M γ is bounded for all < 1. Remark Tm is an important result tat can not be taken for granted. Standard codes will suffer from instabilities wen small parameters, suc as, occur. Note tat a Finite-Element approximation of (2.16 inerits te stability properties of te continuous problem, so one obtains a stability constant tat is independent of te small limit. Let us return to our overall algoritm. Computing an approximate solution, we introduce two errors: One error from using a Finite-Element space instead of te wole Sobolev space, and one from considering ι instead of ι. We start by computing te difference between te latter two: Lemma For a given < 1, let <. Furtermore, let ι and ι be defined as in Sec Its difference can be bounded in terms of and x as ι ι H 1 = O ( 2 x + x 2. (2.44 Proof. From (2.39 and Ass. 2.11, we can conclude tat ι(ϕ ι (ϕ = (u n x ũ n x ϕ 2 (1 (vx n ṽx n ϕ x dx Ω (2.45 C ( + 2 wx n w n x L 2 ϕ H 1 (2.46 = O ( 2 x + x 2 ϕ H 1 (2.47 for a constant C R. Te following lemma bounds te error tat occurs wen using only te approximate rigt-and side ι instead of ι: Lemma For a given < 1, let < < 1. Furtermore, let v n+1 and v n+1 denote te solutions to a(v n+1, ϕ = ι (ϕ ϕ H 1 (Ω, (2.48 a(v n+1, ϕ = ι(ϕ ϕ H 1 (Ω. (2.49 9

11 One can estimate te difference as e 2 = v n+1 v n+1 L 2 ( = O 4 x + 3 x. (2.5 Proof. Te difference between v n+1 and v n+1 can be computed by γ v n+1 v n+1 2 H 1 a(v n+1 v n+1, v n+1 v n+1 (2.51 = ι (v n+1 v n+1 ι(v n+1 v n+1 (2.52 ι ι H 1 v n+1 v n+1 H 1, (2.53 and, subsequently, v n+1 v n+1 L 2 C P F v n+1 v n+1 H 1 (2.54 C ( P F γ ι ι H 1 = O 4 x + 3 x (2.55 because of La and γ 1 = O( 2 for,. 2 Corollary A simple consequence of te proof is tat ( v n+1 v n+1 H 1 = O 4 x + 3 x. (2.56 It is well-known tat, in order to get stable scemes, one needs to link bot and x. In our example, tis can be done in two ways, based on eiter te non-stiff flux f or te total flux f. Let us terefore make te following definition: Definition Te stiff and non-stiff cfl numbers cfl and ĉfl are defined by cfl := x λ max, ĉfl := x λ max, (2.57 respectively. Note tat λ max = 1 and λ max = 1. In our analysis, we rely on te non-stiff cfl number ĉfl, so te cfl number tat is independent on. Let us terefore state te following assumption: Assumption 2.2. We assume tat for a ĉfl R (wic we usually coose to be ĉfl =.8. Remark Tis directly yields = ĉfl x (2.58 e 2 = O ( x. (2.59 Having bounded e 2, we continue by bounding e 3. Lemma Let v n+1 be te Finite-Element solution to (2.31, and let v n+1 be te solution to (2.48. Ten, ( e 3 = v n+1 v n+1 6 L 2 = O x

12 Proof. We are using linear Finite-Elements on a symmetric problem, so one can use te Aubin-Nitsce trick (see, e.g., [6]. As it is crucial for our analysis tat we get te correct dependency of te constant, we perform tis trick ere explicitly. Let us define te dual solution z and its Finite-Element approximation z by One can conclude a(z, ϕ = a(z, ϕ = Ω Ω ( v n+1 v n+1 ϕ dx ϕ H 1 (Ω, ( v n+1 v n+1 ϕ dx ϕ V. v n+1 v n+1 2 L 2 = a(z, vn+1 v n+1 = a(z z, v n+1 v n+1 M z z H 1 v n+1 v n+1 H 1 M x 2 z 2 v n+1 2 C M x2 γ 2 v n+1 v n+1 L 2 ι H 1 = v n+1 v n+1 L 2 ι H 1 O( 4 x v n+1 v n+1 v n+1 v n+1 L 2O L 2O( O( 4 x ( 6 x v 2 denotes te second Sobolev semi-norm, and can be bounded by te rigt-and side of te equation, if te viscosity coefficient is unity. Corollary In a similar way, we can deduce tat ( v n+1 v n+1 6 H 1 = O x x + 2. (2.6 We are now ready to state te final teorem tat assures tat v is approximated consistently. Teorem Let v n+1 be te approximate solution according to te algoritm in Sec. 2.5 wit exact initial data w n w(t n. Under Ass and 2.2, we ave v n+1 v(t n+1 L 2 = O ( x + 6 x 2. (2.61 Proof. We can just collect previous results: v(t n+1 v n+1 L 2 e 1 + e 2 + e 3 (2.62 = O( O ( x ( 6 + O x (2.63 = O ( x + 6 x 2 (

13 Remark Given tat, one can see tat v n+1 is a consistent approximation to v(t n+1, and v(t n+1 v n+1 L 2 = O( 4. By now, we ave sown tat v n+1 is a consistent approximation to v(t n+1. It remains to sow tat also u n+1 is a consistent approximation to u(t n+1. Tereby, u n+1 denotes te function tat is obtained by evaluating (2.11 wit v n+1 instead of v(t n+1. Teorem Let u n+1 be te approximate solution tat is obtained using (2.11 wit v n+1 instead of v(t n+1 ; and wit exact initial data w n w(t n elsewere. Under Ass and 2.2, we ave u n+1 u(t n+1 L 2 = O ( x (2.65 Proof. We can directly compute, exploiting wat we ave already sow: u n+1 u(t n+1 L 2 u n+1 u(t n+1 L 2 + u n+1 u n+1 L 2 (2.66 O( ( v n+1 2 v n+1 x L 2 (2.67 O( 2 + ( 2 v n+1 v n+1 H 1 + v n+1 v n+1 H 1 (2.68 (2.56,(2.6 = O( 2 + O ( 2 + x 2 + O ( 4 x (2.69 = O ( x (2.7 Tere are a few remarks in order: Remark Te solution of te elliptic equation gets more and more difficult wit decreasing time-step, as te elliptic coefficient vanises in tis case. So basically, te metod will only perform well as long as (i.e., for te cfl number of te wole system tere olds cfl, wic is a reasonable assumption. (Oterwise, one would use explicit metods instead. However, coosing = O( 1 p for some p 1, one can observe tat w n+1 w(t n+1 L 2 = O ( 2. Tis directly sows tat te metod works also for te = case. 3. Numerical Results. We compare our sceme wit an Implicit-Euler sceme, and an Implicit/Explicit sceme. Implicit-Euler sceme discretizes w n+1 w n + f(w n+1 = G n+1 (3.1 using a Lax-Friedrics flux. Te Implicit/Explicit sceme proceeds in two steps, discretizing explicitly, and ŵ n w n + f(w n x = G n (3.2 w n+1 ŵ n + f(w n+1 x = (3.3 implicitly, again bot steps wit Lax-Friedrics flux. 12

14 L 2 -Error e-5 1e-6 = 1-1 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme 1e Number of Cells N x L 2 -Error e-5 1e-6 1e-7 1e-8 = 1-2 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme Number of Cells N x L 2 -Error e-6 = 1-4 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme L 2 -Error e-6 = 1-8 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme 1e-8 1e-8 1e Number of Cells N x 1e Number of Cells N x Fig Convergence results for te smoot test case. In dependency on, cfl was set to cfl =.8. Left to rigt, top to bottom: = {1 1, 1 2, 1 4, 1 8 } Smoot test case. As a first, simple test case, we consider a smoot solution on domain Ω = [, 1], given by v(x, t = 2 t sin(2πx (3.4 u(x, t = sin(2πt 2 cos(2πx. (3.5 2π For all metods, we use a (stiff cfl number of cfl =.8. Note tat tis corresponds to a cfl number of ĉfl =.8 wit respect to te non-stiff flux f. If a metod is able to cope wit suc a cfl number, it is called uniformly asymptotically stable. In Fig. 3.2, convergence of te l 2 error at time T =.1 versus number of cells (N x is sown for all tree metods under consideration. One can first observe tat all tree metods are stable for tis unusally large cfl number, as expected. Furtermore, asymptotically (in N x, all metods converge wit order one toward te true solution (u, v, except for te Implicit Euler sceme for = 1 8. We suspect tat tis is because te linear system of equations to be solved in eac time-slab is extremely ill-conditioned. We use Matlab s in-ouse exact solver for linear systems of equations, wic actually yields a corresponding warning. Furtermore, 2 = 1 16 is close to macine zero. Note tat tis does not appen to te Asymptotic Preserving sceme, as its condition number is bounded for. Te really surprising outcome of tis researc is tat te AP sceme performs so muc better tan Implicit Euler and te mixed Implicit / Explicit sceme: Its error is up to four orders of magnitude smaller tan tat of te oter two scemes. We can only suspect tat traditional Finite-Volume scemes do not take advantage of te smoot beaviour of te solution as muc as te Finite-Element metod does Testcase wit a kink. To assess weter te good performance of te asymptotic preserving metod is due to te smootness of te solution, we perform a numerical study on a test case wit a kink, more precisely, we consider again domain 13

15 L 2 -Error e-5 = 1-1 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme L 2 -Error.1 1e-5 1e-6 = 1-2 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme 1e-6 1e-7 1e Number of Cells N x 1e Number of Cells N x L 2 -Error 1e-6 1e-7 1e-8 1e-9 1e-1 1e-11 1e-12 = 1-4 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme 1e Number of Cells N x L 2 -Error.1.1 1e-6 1e-8 1e-1 1e-12 1e-14 1e-16 1e-18 1e-2 = 1-8 Asymptotic Preserving Sceme Implicit / Explicit Sceme Implicit Euler Sceme Number of Cells N x Fig Convergence results for te test case wit a kink. In dependency on, cfl was set to cfl =.8. Left to rigt, top to bottom: = {1 1, 1 2, 1 4, 1 8 }. Ω = [, 1] and te solution v(x, t = 2 t { x x <.5 x + 1 x.5 (3.6 u(x, t = { x 2 2 x <.5 x2 2 + x 1 4 x.5. (3.7 Again, we use a (stiff cfl number of cfl =.8. In Fig. 3.2, convergence of te l2 norm at time T =.1 versus N x is plotted. One can observe tat te scemes converge wit order one up to 1 1, wic is about macine zero (note tat te error as to be scaled wit 2, except for te = 1 8, were Implicit Euler fails to converge for tis large cfl number. For large values of, te scemes nearly perform equally well, wile, for = 1 4, te AP sceme really performs better by orders of magnitude. For = 1 8, bot te AP and Implicit / Explicit sceme perform about equally well. Neverteless, as 2 = 1 16 is close to macine zero, tese results are not too reliable. 4. Conclusions and Outlook. We ave compared te recently developed AP scemes versus more traditional Finite-Volume scemes for te p system. It was demonstrated tat te AP scemes outperform bot Implicit Euler and an Implicit / Explicit sceme by orders of magnitude if tere is a small parameter. We are interested in te use of ig-order metods, also in te context of asymptotic preserving scemes. In particular, our interest lies in te use of Discontinuous Galerkin metod [1, 9, 8, 7, 11, 4, 16, 1, 14, 2]. Future work will terefore treat an asymptotic preserving discontinuous Galerkin sceme applied to (1.1-(1.2 for various orders of consistency, and also compare performance of te AP scemes versus Diagonally-Implicit-Runge-Kutta (DIRK. It is to be expected tat te ig order of consistency will reduce te effect tat we could observe in tis publication. Neverte- 14

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