Stability of the fourth order Runge-Kutta method for time-dependent partial. differential equations 1. Abstract

Transcription

1 Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations 1 Zheng Sun 2 and Chi-Wang Shu 3 Abstract In this paper, we analyze the stability of the fourth order Runge-Kutta method for integrating semi-discrete approximations of time-dependent partial differential equations. Our study focuses on linear problems and covers general semi-bounded spatial discretizations. A counter example is given to show that the classical four-stage fourth order Runge-Kutta method can not preserve the one-step strong stability, even though the ordinary differential equation system is energy-decaying. But with an energy argument, we show that the strong stability property holds in two steps under an appropriate time step constraint. Based on this fact, the stability extends to general well-posed linear systems. As an application, we utilize the results to examine the stability of the fourth order Runge-Kutta approximations of several specific method of lines schemes for hyperbolic problems, including the spectral Galerkin method and the discontinuous Galerkin method. Keywords: stability analysis, Runge-Kutta method, method of lines, spectral Galerkin method, discontinuous Galerkin method. 1 Research is supported by ARO grant W911NF and NSF grant DMS Division of Applied Mathematics, Brown University, Providence, RI zheng sun@brown.edu 3 Division of Applied Mathematics, Brown University, Providence, RI shu@dam.brown.edu. 1

2 1 Introduction In common practice, the method of lines for solving time-dependent partial differential equations (PDEs) starts with a spatial discretization to reduce the problem to an ordinary differential equation (ODE) system solely dependent on time, then a suitable ODE solver is used for the time integration. The issue of stability arises from such a procedure, whether a stable semi-discrete scheme will still be stable after coupling with the time discretization. In this paper, we focus on the stability of the fourth order Runge-Kutta (RK) method in this context. The model problem for stability analysis is usually chosen as a well-posed linear system t u = L(x, t, x )u. But firstly, let us consider a simpler problem t u = L(x, x )u with L + L 0. (Here the superscript stands for the adjoint, to be distinguished from T, which stands for the transpose. But for matrices, they are the same under the usual dot product in R m.) Its semi-discrete scheme corresponds to an autonomous ODE system d dt u N(t) = L N u N (t). Here L N is a constant matrix and the parameter N relates with the degree of freedom in the spatial discretization. Suppose L N inherits the semi-negativity L N + L N 0. Then d u dt N(t) 2 = (L N u N, u N ) + (u N, L N u N ) 0, where we temporarily assume (, ) to be the usual dot product and to be the induced norm. Therefore the solution to the semidiscrete scheme is strongly stable. We are interested in whether this strong stability will be preserved after the four-stage fourth order RK time discretization. In other words, with τ being the time step and u n+1 N = P 4(τL N )u n N = (I + τl N + 12 (τl N) (τl N) ) (τl N) 4 u n N, we wonder whether u n+1 N un N, or equivalently, whether the operator norm of P 4(τL N ) is bounded by 1. If this is true, the stability extends to general semi-bounded systems t u = L(x, t, x )u by using a standard argument. 2

3 For L N being a scalar and τ being sufficiently small, the proposition above can be justified by analyzing the region of absolute stability, see Chapter IV.2 in [15], for example. A natural attempt is to apply the eigenvalue analysis to extend the result to systems. However, this technique can only be used for normal matrices L N, namely L N L N = L N L N. If renorming is allowed, the method also covers diagonalizable L N. (But it would still be useless if the diagonalizing matrix is ill-conditioned, see [10].) While for general systems, the naive eigenvalue analysis may end up with misleading results. We refer to Chapter 17.1 in [8] for a specific example. The analysis for the general systems initiates from the coercive problems, namely L N + L N ηl N L N for some positive constant η. In [10], Levy and Tadmor used the energy method to prove the strong stability of the third order and the fourth order RK schemes under the coercivity condition and the time step constraint τ cη, where c = 3 50 third order scheme and c = 1 62 for the for the fourth order scheme. Later in [6] and [14], a simpler proof was provided, with the time step constraint been significantly relaxed to c = 1, which extends to general s-stage s-th order linear Runge-Kutta schemes. The authors in [6] pointed out that the coercivity condition implies the strong stability of the forward Euler scheme when τ η. The strong stability of the high order RK schemes follows from the fact that they can be rewritten as convex combinations of the forward Euler steps. These results coincides with the earlier work on contractivity analysis of the numerical solutions to ODE systems, see [12] and [9]. In their theory for contractivity, or strong stability in our context, a circle condition is assumed, which is essentially equivalent to the strong stability assumption for the forward Euler scheme in [6]. In general, the coercivity condition does not hold for method of lines schemes arising from purely hyperbolic problems. Hence it is still imperative to analyze schemes which only satisfy a semi-negativity condition L N + L N 0. In [14], for the third order RK scheme, Tadmor successfully removed the coercivity assumption and proved that the third order RK scheme is strongly stable if L N + L N 0 and τ L N 1. But the strong stability for the 3

4 fourth order RK method remains open, which is the main issue we are concerned with in this paper. We find that, although the strong stability of the fourth order RK method passes the examination of the scalar equation, and can be extended to normal systems, it does NOT hold in general. More specifically, we have the following counter example in Proposition 1.1, whose proof is given in Appendix A Proposition 1.1. Let L N = Then L N + L T N 0 and P 4(τL N ) > 1 for τ sufficiently small. Interestingly, despite the negative result on the one-step performance, the strong stability property holds in two time steps. In other words, with the constraint τ L N c 0 for some positive constant c 0, P 4 (τl N ) 2 1 and hence u n+2 N un N. This is still sufficient to justify the stability after long time integration. It can also be used to obtain the stability u n N K(tn ) u 0 N for semi-bounded and time-dependent L N. We will apply the results to study the stability of different spatial discretizations coupled with the fourth order RK approximation. The paper is organized as follows. In Section 2 we exhibit our main results, and prove the stability of the fourth order RK method with skew-symmetric, semi-negative and semibounded L N. The case for L N dependent on time is also discussed. In Section 3, we apply our results to several different spatial discretizations. We specifically focus on Galerkin methods as a complementary of [14], including the spectral Galerkin method and the discontinuous Galerkin method. Finally, we give our concluding remark in Section 4. 2 Stability of the fourth order RK method In this section, we analyze the stability of the fourth order RK schemes. For simplicity, we assume everything to be real, but the approach extends to the complex spaces. To facilitate our later discussion on applications to spatial discretizations based on Galerkin methods, 4

5 we discuss over a general Hilbert space. We avoid clearly characterizing the method of lines scheme in this general setting. But it will not lead to ambiguities, since it only serves as a motivation to derive the fourth order RK iteration, and will not be used in the stability analysis. To reduce to ODE systems, one simply sets V = R m and (, ) to be the usual dot product. We also remark that, by using the inner product (u, v) H = u T Hv with H being a symmetric positive definite matrix, L N + L N 0 is equivalent to LT N H + HL N 0, which is consistent with the assumption in equation (6) of [14]. 2.1 Notations and the main results We denote by V a real Hilbert space equipped with the inner product (, ). The induced norm is defined as = (, ). Consider a method of lines scheme defined on V. t u N = L N u N, (2.1) where u N (, t) V and L N is a bounded linear operator on V. Suppose L N is independent of t, then the four-stage fourth order RK approximation can be written as u n+1 N = P 4(τL N )u n N, P 4(τL N ) = I + τl N (τl N) (τl N) (τl N) 4. (2.2) When L N = L N (t) depends on time, we denote by and the specific form of R τ (t) will be introduced latter. u n+1 N = R τ(t)u n N (2.3) For simplicity, we drop all the subscripts N in the remaining parts of the section. Given an operator L, the operator norm is defined as L = sup v =1 Lv. We denote by B(V) = {L : L < + } the collection of bounded linear operators on V. For any L B(V), there exists a unique L B(V) such that (Lw, v) = (w, L v). L is referred as the adjoint of L. If (v, (L + L )v) 0, v V, denoted by L + L 0, then we say L is semi-negative. A special case is when L + L = 0, L is referred to be skew-symmetric. If L + L 2µI for some positive number µ, namely L µi is semi-negative, then we say L is semi-bounded. 5

6 We denote by [w, v] = (w, (L+L )v) and w = [w, w]. [w, v] is a bilinear form on V. For L + L 0, [w, v] is a semi-inner-product, and one has the Cauchy-Schwarz inequality, [w, v] w v. To simplify our notation, we define L = τl. This notation will be used in the intermediate lemmas and the proofs. For clearness, we restore the notation τl in our main theorems. Here we summarize our main results on the stability of the fourth order RK approximation. c 0 and K are some positive real numbers in the following statements. The details will be given in Theorem 2.1, Corollary 2.1, Theorem 2.2, Theorem 2.3 and Theorem 2.4. (1) Suppose L + L = 0. (2.2) is strongly stable, u n+1 u n, if τ L 2 2. (2a) Suppose L+L 0, LL = L L. (2.2) is strongly stable, u n+1 u n, if τ L c 0. (2b) Suppose L+L 0. (2.2) is strongly stable in two steps, u n+2 u n, if τ L c 0. (3) Suppose L + L 2µI. (2.2) is stable, u n K(t n ) u 0, if τ c 0 L +µ. (4) Suppose L = L(t) satisfies certain regularity assumptions, and L + L 2µI. Then (2.3) is stable, u n K(t n ) u 0, under an appropriate time step constraint. Here the results with strong assumptions are listed first. But one should note that (2b) is the most crucial one among them. (1) is obtained from an intermediate step in deriving (2b), while (2a), (3) and (4) are all corollaries of (2b). 2.2 Strong stability for skew-symmetric L The first thing we would do is to derive an energy equality, which would be useful in understanding the subtle change of the norm of the solution after one time step. To this end, we introduce several identities in Lemma 2.1, and then use them to derive the equality in Lemma 2.2. Lemma 2.1. (Lv, v) = τ 2 v 2, (2.4) (L 2 v, v) = Lv 2 τ[lv, v], (2.5) (L 3 v, v) = τ 2 Lv 2 τ[l 2 v, v], (2.6) 6

7 (L 4 v, v) = L 2 v 2 + τ[l 2 v, Lv] τ[l 3 v, v]. (2.7) Proof. (1) (Lv, v) = 1 (Lv, v) + 1 (v, Lv) = 1 (v, (L L )v) = τ 2 v 2. (2) (L 2 v, v) = (Lv, L v) = (Lv, Lv) + (Lv, (L + L )v) = Lv 2 τ[lv, v]. (3) (L 3 v, v) = (L 2 v, L v) = (L(Lv), Lv) + (L 2 v, (L + L )v) = τ 2 Lv 2 τ[l 2 v, v], where we have used (2.4) in the last equality. (4) (L 4 v, v) = (L 3 v, L v) = (L 2 (Lv), Lv) + (L 3 v, (L + L )v) = L 2 v 2 + τ[l 2 v, Lv] τ[l 3 v, v], where (2.5) is used in the last equality. Lemma 2.2 (Energy equality). u n+1 2 u n 2 = Q 1 (u n ), where and Q 1 (u n ) = L4 u n L3 u n 2 + τ A = (α ij ) 4 4 = 3 α ij [L i u n, L j u n ], (2.8) i,j=0 1 1/2 1/6 1/24 1/2 1/3 1/8 1/24 1/6 1/8 1/24 1/48 1/24 1/24 1/48 1/144 Proof. Taking an inner product of (2.2) with u n, one has. 1 2 un un un+1 u n 2 = ((L L L L4 )u n, u n ). Applying (2.4) - (2.7) with v = u n, we obtain u n+1 2 u n 2 = u n+1 u n 2 Lu n L2 u n 2 τ u n 2 τ[lu n, u n ] + τ 6 Lun 2 τ 3 [L2 u n, u n ] + τ 12 [L2 u n, Lu n ] τ 12 [L3 u n, u n ]. (2.9) Using (2.4) - (2.6) with v = Lu n, the first two terms on the right can be expanded. u n+1 u n 2 Lu n 2 = (u n+1 u n Lu n, u n+1 u n + Lu n ) = u n+1 u n Lu n 2 + (( 1 2 L L L4 )u n, 2Lu n ) = u n+1 u n Lu n L2 u n 2 τ 2 Lun 2 (2.10) τ 3 [L2 u n, Lu n ] + τ 24 L2 u n 2 τ 12 [L3 u n, Lu n ]. 7

8 Substitute (2.10) into (2.9), one has u n+1 2 u n 2 = u n+1 u n Lu n L2 u n 2 τ u n 2 τ[lu n, u n ] τ 3 Lun 2 τ 3 [L2 u n, u n ] τ 4 [L2 u n, Lu n ] (2.11) τ 12 [L3 u n, u n ] + τ 24 L2 u n 2 τ 12 [L3 u n, Lu n ]. Similar as before, one can use (2.4) and (2.5) with v = L 2 u n to calculate the difference of square terms on the right. u n+1 u n Lu n L2 u n 2 =(u n+1 u n Lu n 1 2 L2 u n, u n+1 u n Lu n L2 u n ) = u n+1 u n Lu n 1 2 L2 u n 2 + (( 1 6 L L4 )u n, L 2 u n ) = u n+1 u n Lu n 1 2 L2 u n L3 u n 2 τ 12 L2 u n 2 τ 24 [L3 u n, L 2 u n ]. (2.12) Once again, we plug (2.12) into (2.11) to get u n+1 2 u n 2 = u n+1 u n Lu n 1 2 L2 u n L3 u n 2 Finally, note that Therefore τ u n 2 τ[lu n, u n ] τ 3 Lun 2 τ 3 [L2 u n, u n ] τ 4 [L2 u n, Lu n ] τ 12 [L3 u n, u n ] τ 24 L2 u n 2 τ 12 [L3 u n, Lu n ] τ 24 [L3 u n, L 2 u n ]. u n+1 u n Lu n 1 2 L2 u n L3 u n 2 = u n+1 u n Lu n 1 2 L2 u n 1 6 L3 u n 2 + ( 1 24 L4 u n, 1 3 L3 u n ) = L4 u n 2 τ 144 L3 u n 2. Q 1 (u n ) = L4 u n L3 u n 2 τ u n 2 τ[lu n, u n ] τ 3 Lun 2 which can be rewritten as (2.8). τ 3 [L2 u n, u n ] τ 4 [L2 u n, Lu n ] τ 12 [L3 u n, u n ] τ 24 L2 u n 2 τ 12 [L3 u n, Lu n ] τ 24 [L3 u n, L 2 u n ] τ 144 L3 u n 2, 8

9 According to Lemma 2.2, the energy change Q 1 (u n ) consists of two parts, the numerical dissipation L4 u n Lun 2 and the generalized quadratic form τ 4 i,j=0 α ij[lu i, Lu j ]. When L is skew-symmetric, the quadratic form is simply 0. Hence one can obtain the onestep strong stability. Theorem 2.1. Suppose L + L = 0, then the fourth order RK approximation (2.2) to the method of lines scheme (2.1) is strongly stable, u n+1 u n, if τ L 2 2. Proof. L + L = 0 implies [w, v] = 0, v, w. Hence when τ L 2 2, u n+1 2 u n 2 = Q 1 (u n ) = L4 u n L3 u n 2 ( L ) L3 u n Stability for semi-negative L According to Lemma 2.2, for non-skew-symmetric L, we would need to absorb the quadratic form with the help of the numerical dissipation. One can see that the high order terms L i u n with i 3 are easy to control. But there are no other terms to bound u n, Lu n and L 2 u n. Our only hope is that τ 2 ij=0 α ij[lu i, Lu j ] itself is negative-definite. Lemma 2.3 indicates that, to check the negativity of the generalized quadratic form, one only needs to examine the coefficient matrix, as that for the polynomials. In Lemma 2.4, we prove a sufficient condition for strong stability, once 2 ij=0 α ij[lu i, Lu j ] is negative-definite, the energy change will be non-positive when L is sufficiently small. Lemma 2.3. Suppose L is semi-negative. Let M = (m ij ) be a real symmetric semi-negative definite matrix, then i,j m ij[u i, u j ] 0. Proof. Suppose M = S T ΛS, where S = (s ij ) is an orthogonal matrix and Λ is a diagonal matrix with the diagonal elements λ i 0. Note that m ij = k λ ks ki s kj. Therefore m ij [u i, u j ] = ( λ k s ki s kj )[u i, u j ] = λ k ( s ki s kj [u i, u j ]) i,j i,j k k i,j = k λ k [ i s ki u i, j s kj u j ]) = k λ k i s ki u i

10 Lemma 2.4. Suppose L is semi-negative. Let Q(u) = α L 3 u 2 + τ a α ij [L i u, L j u], α 00 α 01 α 02 where α ij = α ji and a 2. If α < 0 and A 2 = α 10 α 11 α 12 is negative-definite, then α 20 α 21 α 22 there exists c 0 > 0, such that Q(u) 0 as long as L c 0. Proof. Let ε be the largest eigenvalue of the matrix A 2. Then A 2 + εi is semi-negative definite. According to Lemma 2.3, 2 2 α ij [L i u, L j u] = (α ij + εδ ij )[L i u, L j u] ε( u 2 + Lu 2 + L 2 u 2 ) i,j=0 i,j=0 i,j=0 ε( u 2 + Lu 2 + L 2 u 2 ) By Cauchy-Schwartz and the arithmetic-geometric mean inequality, a a α ij [L i u, L j u] ε( u 2 + Lu 2 + L 2 u 2 ) + α i L i u 2 max(i,j)>2 ε( u 2 + Lu 2 + L 2 u 2 ) + i=3 a 2 α i L L 2(i 3) L 3 u 2, i=3 (2.13) (2.14) where we have used the fact v 2 2 L v 2 in the last inequality, and α i are some nonnegative constants depending on ε and α ij. Using (2.13) and (2.14), if L = τ L c 0, one has Q(u) (α + a i=3 2 α i c 2(i 3)+1 0 ) L 3 u 2. Since α is negative, Q(u) is non-positive as long as c 0 is sufficiently small. Unfortunately, Q 1 (u n ) does not satisfy the assumptions in Lemma 2.4. Actually, the 1 1/2 1/6 three eigenvalues of A 2 = 1/2 1/3 1/8 are , and /6 1/8 1/24 This motivates us to disprove the strong stability of the fourth order RK method and we end up with the counter example in Proposition 1.1. So we turn to seek the power-boundedness of P 4 (L). Surprisingly, though Q 1 (u n ) itself fails to pass Lemma 2.4, Q 1 (u n+1 ) + Q 1 (u n ) succeeds. Hence we obtain the two-step strong stability in Theorem

11 Theorem 2.2 (Two-step strong stability for semi-negative L). Suppose L+L 0, then the fourth order RK approximation (2.2) to the method of lines scheme (2.1) is strongly stable in two steps, u n+2 u n, if τ L c 0. Proof. Let u n+2 = P 4 (L)u n+1. Then u n+2 2 u n+1 2 = Q 1 (u n+1 ). Substitute (2.2) into Q 1 (u n+1 ) and rewrite the quadratic form in terms of u n. By direct calculation, one can obtain where While where When L 2, Hence where Q 1 (u n+1 ) = L4 u n L3 u n τ 7 i,j=0 1 3/2 7/6 Ã 2 = ( α ij ) 3 3 = 3/2 7/3 15/8. 7/6 15/8 37/24 Q 1 (u n ) = L4 u n L3 u n 2 + τ α ij [L i u n, L j u n ], 3 ˆα ij [L i u n, L j u n ], i,j=0 1 1/2 1/6 Â 2 = (ˆα ij ) 3 3 = 1/2 1/3 1/8. 1/6 1/8 1/ L4 u n L3 u n+1 2 0, L4 u n L3 u n L3 u n 2. u n+2 2 u n 2 = Q 1 (u n+1 ) + Q 1 (u n ) L3 u n 2 + τ 7 α ij [L i u n, L j u n ], i,j= /3 A 2 = (α ij ) 3 3 = Ã2 + Â2 = 2 8/3 2. 4/3 2 19/12 The three eigenvalues of A 2 are , and Hence A 2 is negative definite, and one could complete the proof by applying Lemma

12 Remark 2.1. There is another interpretation of the two-step strong stability. We treat u n+1 as a intermediate stage and consider u n+2 as the solution of an eight-stage fourth order RK scheme composed by two four-stage fourth order RK time stepping. Then Theorem 2.2 states this new RK scheme is strongly stable. This coincides with Remark 3 of Section 4 in [14], where the author conjectured the fourth order RK scheme can preserve the strong stability if allowing additional stages. Remark 2.2. Here we only focus on the existence of the constant c 0 instead of giving a specific estimate. The argument above is far from sharp. To obtain a reasonable bound for c 0, one should expand L4 u n L3 u n+1 2 to separate the quadratic form involving [, ], and carefully go through all the constants. To this end, a generalized version of Lemma 2.1 is also needed. We leave these to interested readers. Realizing the fact that, for a normal operator G, G 2 1 implies G 1, we prove the one-step strong stability for normal and semi-negative L. Corollary 2.1 (Strong stability for normal and semi-negative L). Suppose LL = L L and L + L 0, then the fourth order RK approximation (2.2) to the method of lines scheme (2.1) is strongly stable, u n+1 u n, if τ L c 0 for some constant c 0. Proof. According to Theorem 2.2, we have P 4 (L) 2 1 if L c 0. When L is normal, P 4 (L) is also normal, P 4 (L)P 4 (L) = P 4 (L) P 4 (L). Hence for any u V, P 4 (L)u 2 = (P 4 (L)u, P 4 (L)u) = (u, P 4 (L) P 4 (L)u) = (u, P 4 (L)P 4 (L) u) = P 4 (L) u 2. Therefore, under the same constraints, L c 0, one has P 4 (L) 2 = sup P 4 (L)u 2 = sup (u, P 4 (L) P 4 (L)u) u =1 u =1 sup P 4 (L) P 4 (L)u = sup P 4 (L) 2 u = P 4 (L) 2 1. u =1 u =1 In other words, u n+1 u n if L c 0. Remark 2.3. By using scalar eigenvalue analysis, the sharp bound of c 0 in Corollary 2.1 is 2 2, as that in Theorem

13 2.4 Stability for semi-bounded L We then apply the perturbation analysis to obtain the stability for semi-bounded L. In the following proof, one actually needs to assume the time steps satisfies τ 2k = τ 2k+1 in order to apply Theorem 2.2. For simplicity, we use a uniform time stepping, namely τ k = τ. The same assumption will be used in Section 2.5. Theorem 2.3. Suppose L+L 2µI, then the fourth order RK approximation (2.2) to the method of lines scheme (2.1) is stable, u n K(t n ) u 0, if τ c 0 L +µ c 0. Here K(t n ) is a constant depending on µ and t n. Proof. Note that for some constant P 4 (τl) 2 = P 4 (τ(l µi) + τµi) 2 = P 4 (τ(l µi)) 2 + τµp(τµ, τ(l µi)), where P(s, G) is of the form P(s, G) = i,j 0, i+j 7 α ijs i G j. Since L + L 2µI, (L µi) + (L µi) 0. According to Theorem 2.2, P 4 (τ(l µi)) 2 1 if τ L µi c 0. Under a stricter restriction τ c 0, one also has L +µ P(τµ, τ(l µi)) i,j 0, i+j 7 α ij τµ i τ(l µi) j i,j 0, i+j 7 α ij c i+j 0. Therefore for τ c 0 L +µ, we have P 4(τL) cµτ, where c = i,j 0, i+j 7 α ij c i+j 0. With this one-step estimate, one has u 2k P 4 (τl) 2 k u 0 (1 + cµτ) t2k 2τ u 0 K(t 2k ) u 0, and u 2k+1 P 4 (τl) u 2k P 4 (c 0 )K(t 2k ) u 0, which prove the stability. 2.5 Stability for semi-bounded and time-dependent L We conclude this section by extending the results to L dependent on time. One should note in this case, the fourth order RK scheme can no longer be written as the truncated exponential 13

14 in (2.2). Also different fourth order RK time integrators are no longer equivalent. One can use the classical four-stage fourth order RK scheme in (2.15) as an example for our stability analysis. However, our proof does not rely on this specific form and can be used for general cases. The classical four-stage fourth order RK scheme is k 1 = L(t n )u n, k 2 = L(t n+1 2 )(u n + τ 2 k 1), k 3 = L(t n+1 2 )(u n + τ 2 k 2), (2.15) k 4 = L(t n+1 )(u n + τk 3 ), u n+1 = u n + τ 6 (k 1 + 2k 2 + 2k 3 + k 4 ), where t n+1 2 = t n + τ. As a short hand notation, let us denote by 2 u n+1 = R τ (t n )u n, where R τ (t n ) =I + τ ( ) L(t n ) + 4L(t n+1 2 ) + L(t n+1 ) 6 ( ) + τ2 L(t n )L(t n+1 2 ) + L(t n+ 1 2 ) 2 + L(t n+1 2 )L(t n+1 ) 6 ( ) + τ3 L(t n )L(t n+1 2 ) 2 + L(t n+1 2 ) 2 L(t n+1 ) L(tn )L(t n+ 1 2 ) 2 L(t n+1 ). + τ4 The two-step approximation can be written as u n+2 = R τ (t n+1 )R τ (t n )u n. (2.16) We also assume that L satisfies the Lipschitz continuity condition L(t 2 ) L(t 1 ) η t 2 t 1, (2.17) with sup t L(t) η < + for some constant η. Lemma 2.5. Suppose L satisfies (2.17). Then m L(t + α i τ) (L(t)) m m α i η m τ. 14

15 Proof. We prove by induction. According to assumption (2.17), the lemma holds for m = 1. Suppose it holds for m, then L(t + α i τ) (L(t)) m+1 m+1 m+1 m m L(t + α i τ) L(t) L(t + α i τ) + L(t) L(t + α i τ) (L(t)) m+1 m m L(t + α m+1 τ) L(t) L(t + α i τ) + L(t) L(t + α i τ) (L(t)) m α m+1 ητ η m + η ( α i η m+1 τ, m+1 = m α i )η m τ where we have used the inductive assumption and η > sup t L(t) in the second last line. Hence the lemma holds for any positive integer m. Theorem 2.4. Suppose L = L(t) and L+L 2µI for some positive number µ. L satisfies the Lipschitz continuity condition (2.17). Then the four-stage fourth order RK approximation (2.15) to the method of lines scheme (2.1) is stable, u n K(t n ) u 0, if τ c 0 η+µ. Here K(t n ) is some constant depending on µ and t n. Proof. And P 4 (τl) 2 = I + 2τL + 2(τL) (τl) (τl) (τl) (τl) (τl) (τl)8. R τ (t n+1 )R τ (t n ) = 8 i=0 j α ij (2.18) i (τl(t n + α ijk τ)), (2.19) with { j α ij} 8 i=0 take values of 1, 2, 2, 4 3, 2 3, 1 4, 5 72, 1 72 and as those in P 4(τL) 2. (This must be true since when L is independent of time, (2.19) reduces to (2.18).) Applying Lemma k=1 15

16 2.5, one has 8 i R τ (t n+1 )R τ (t n ) P 4 (τl) 2 = α ij τ i ( L(t + α ijk τ) (L(t)) i ) i=0 j 8 α ij τ i τ i=0 j 8 α ij ( i=0 j k k=1 i L(t + α ijk τ) (L(t)) i k=1 i α ijk )(τη) i = cτ, On the other hand, P 4 (τl) cµτ if τ(η + µ) c 0, hence R τ (t n+1 )R τ (t n ) P 4 (τl) 2 + R τ (t n+1 )R τ (t n ) P 4 (τl) (cµ + c)τ. As we have done in Theorem 2.3, one can prove u n K(t n ) u 0, if τ c 0 η+µ. Remark 2.4. For different four-stage fourth order RK schemes, the values of α ij and α ijk are different. But j α ij are always the same. Hence the proof above would still work. 3 Applications to hyperbolic problems In this section, we apply the previous results to several semi-discrete approximations of hyperbolic problems, and justify their stability after coupling with the fourth order RK time integrator. The time step restriction is also referred as the CFL condition in this context. In [14], Tadmor has provided detailed examples for spatial discretization based on the nodal-value formulation, including the finite difference method and spectral collocation method. Complementary to [14], we will mainly focus on Galerkin methods, including the global approach, the spectral Galerkin method, and the local approach, the finite element discontinuous Galerkin method. The Galerkin methods are spatial discretization techniques based on the weak formulation. Consider the initial value problem, t u(x, t) = L(x, t, x )u(x, t), (x, t) Ω (0, T), u(x, 0) = u 0 (x), x Ω. 16

17 The Galerkin methods seeks a solution of the form u N (x, t) = N k=0 c k(t)φ k (x). Here {φ k (x)} N k=0 forms the basis of a finite dimensional space V on Ω, equipped with the inner product (, ). In addition, one requires that ( t u N, v N ) = B(u N, v N ), v N V, u N (x, 0) = Pu 0. Here B is a bilinear form on V with B(u N, v N ) approximating (Lu N, v N ), and P is the projection to V. For spectral Galerkin method, V is chosen as the span of the trigonometric functions or polynomials over the whole domain. Since the functions in V are sufficiently smooth. One can directly set B(u N, v N ) = (Lu N, v N ). Hence the method can be written as t u N = L N u N = PLu N = PLPu N. It suffices to check whether L N = PLP satisfies the conditions in Section 2 and examine L N to obtain the time step constraint. For discontinuous Galerkin method, V is a piecewise polynomial space based on an appropriate partition of the domain Ω. The associated bilinear form is involved with the so-called numerical flux. We will explain it in the latter part of the section. 3.1 Spectral Galerkin method The following examples are based on Example 3.8 and Example 8.3, 8.4 in [7]. The estimate of L N relies on the inverse inequalities on the finite dimensional trigonometric or polynomial spaces, which we refer to [11] for details. For interested readers, we also refer to the same books [7] and [11] for a systematic set up of the spectral methods Fourier method Consider the linear advection equation u t = a(x, t)u x, t (0, T), x (0, 2π), (3.1) 17

18 with periodic boundary conditions. We assume that a(x, t) is a smooth function and is periodic with respect to x. Under this circumstance, L = a(x, t) x. One has (v, (L + L )v) = (a x v, v) + (v, a x v) = (v, x (av)) + (v, a x v) = (v, a x v) sup a x v 2, x,t where (, ) is specified as the L 2 inner product on [0, 2π]. Hence L + L sup x,t a x I. In the Fourier Galerkin method, we set V = span{sin(kx), cos(kx)} N k=0, and choose (, ) to be the L 2 inner product on [0, 2π] as well. Note that P is self-adjoint, hence The semi-boundedness holds. L N + L N = PLP + (PLP) = P(L + L )P sup a x I. x,t On the other hand, by using the inverse inequality for trigonometric functions φ N φ, φ V and the fact P 1, one has and L N a(x, t) x sup a N, x,t L N (t 2 ) L N (t 1 ) (a(, t 2 ) a(, t 1 )) x sup a t N t 2 t 1. x,t Hence, the Lipschitz condition holds for η = N max{sup x,t a, sup x,t a t }. We obtain the following corollary of Theorem 2.4. Corollary 3.1. Consider the Fourier Galerkin approximation of the advection equation (3.1) with the fourth-order RK time discretization, u n+1 N = R τ(t n )u n N, R τ (t n ) defined in (2.16), with L N = P(a(x, t) x )P. This fully-discrete scheme is stable, u n N K(t n ) u 0 N, under the CFL condition τ(n max{sup x,t a, sup x,t a t } sup x,t a x ) c 0, for some constant c 0. 18

19 3.1.2 Polynomial method Consider the linear advection equation t u = x u, t (0, T), x ( 1, 1), (3.2) with the inflow boundary condition u(1, t) = 0. In the polynomial Galerkin method, we set V = {φ(x) span{x k } n k=0 φ(1) = 0} and (, ) w is the weighted L 2 inner product, namely (f, g) w = 1 w(x)f(x)g(x)dx. The corre- 1 sponding norm is denoted as w. We denote by P w the projection to V under (, ) w. Different weight functions w(x) correspond to different polynomial methods in the literature. Let us consider the Jacobi method as an example. Here w(x) = (1+x) α (1 x) β with α 0 and 1 < β 0. When α = β = 0, w(x) = 1, and the method is also referred as the Legendre method. =2 It can be shown that L N is semi-negative in this setting. For any u N V, one has (u N, (L N + L N)u N ) w = (P w u N, LP w u N ) w + (LP w u N, P w u N ) w = 2(u N, Lu N ) w 1 1 wu N x u N dx = For α > 0 and β < 0, w x u 2 N dx = w u 2 N dx w( 1)u N( 1, t) 2 1 w (x) = α(1 + x) α 1 (1 x) β β(1 + x) α (1 x) β 1 = (1 + x) α 1 (1 x) β 1 (α(1 x) β(1 + x)) 0. Similarly, one can prove w (x) 0 for other cases. Hence L N + L N w u 2 N dx. Furthermore, we apply the inverse inequality for Jacobi polynomials to obtain L N w CN 2 for some constant C. (Refer to Theorem 3.34 in [11].) Therefore, by applying Theorem 2.2, one has the following corollary. Corollary 3.2. Consider the Jacobi Galerkin approximation of the advection equation (3.2) with the fourth-order RK time discretization, u n+1 N = P 4(τL N )u n N, L N = P w ( x )P w. 19

20 This fully-discrete scheme is strongly stable in two steps, u n+2 N w u n N w, under the CFL condition τ CN 2, where C depends on the constant in the inverse inequality. 3.2 Discontinuous Galerkin method A detailed discussion of the example in Section can be found in [13]. For a systematic set up of the discontinuous Galerkin method for solving conservation laws, one can refer to a series of paper [4], [3], [2], [1] and [5] by Cockburn et al. To be consistent with the past literatures, we switch the subscript N to h in this section, where h corresponds to the mesh size. Also, we will use the bold fonts for vectors and matrices in this section Multi-dimensional scalar equation Consider the linear scalar conservation law t u(x, t) = (β(x)u(x, t)), x Ω R d, t (0, T), (3.3) with the periodic boundary conditions. Here β is a smooth function satisfying the divergencefree condition, β(x) = 0. Suppose K = {K} is a quasi-uniform partition of the domain Ω. We denote by h the largest diameter of the elements. The collection of cell interfaces is denoted by E. Let us define V = {v L 2 (Ω) : v K P p (K), K K}, where P p (K) is the space of polynomials of degree no more than p on K. As for (, ), we use the L 2 inner product on Ω. In discontinuous Galerkin method, one seeks a solution satisfying Ω t u h v h dx = K ( u h (βv h )dx + û h v h β ndl), v h V, K K where û h is the numerical flux, approximating the trace of u along the edges. A popular choice is to obey the unwinding principle. In our case, that is û h = u + h, where u+ h = lim ε 0 + u h (x+εβ). 20

21 We also use u h = lim ε 0 + u h (x εβ) to represent the trace of u h from the downwind side. When β(x) is parallel to the cell interfaces, u ± h are not well-defined. But in this case, β(x) n = 0, hence the value of u ± h will not make any difference. Let us introduce the short hand notation, H + β (w, v) = ( w (βv)dx K K K Then the scheme becomes, find u h V, such that ( t u h, v h ) = H + β (u h, v h ), v h V. K w + vβ ndl ). (3.4) H + β has the following properties. We refer to [13] for details. Lemma 3.1. For any w h, v h V, we have (1) H + β (v h, v h ) = 1 v 2 h 2 β, where v h β = e E e (v+ h v h )2 β n dl. (2) H + β (w h, v h ) Ch 1 w h v h, for some constant C depending on β and the constant in the inverse estimate. By defining (L h u h, v h ) = H + β (u h, v h ), (3.5) we can also rewrite the scheme as t u h = L h u h. Using Lemma 3.1, one has (v h, (L h + L h )v h) = 2(L h v h, v h ) = 2H + β (v h, v h ) = v h 2 β 0, (3.6) L h v h 2 = H + β (v h, L h v h ) Ch 1 v h L h v h. (3.7) Hence L h is semi-negative and L h Ch 1. The stability of the fully discretized scheme now follows as a corollary of Theorem 2.2. Corollary 3.3. Consider the discontinuous Galerkin approximation of (3.3) with the fourthorder RK time discretization, u n+1 h = P 4 (τl h )u n h, L h defined in (3.4) and (3.5). 21

22 This fully-discrete scheme is strongly stable in two steps, u n+2 h u n h, under the CFL condition τ Ch, where C depends on β and the constant in the inverse estimate Multi-dimensional system We now consider the symmetric hyperbolic system t u(x, t) = d A i xi u(x, t), x = (x 1,..., x d ) Ω R d, u = (u 1,..., u m ) T R m, (3.8) where A i are m m constant real symmetric matrices. For simplicity, we assume Ω to be a hypercube in R d and apply the periodic boundary conditions. Again, we denote by K = {K} a quasi-uniform partition of the domain Ω with the mesh size h. And E is the collection of the cell interfaces. The space V is chosen as V = {v [L 2 (Ω)] m v = (v 1,..., v m ) T, v k K P p (K), K K}, with the inner product (w, v) = K K (w, v) K = K K K w vdx and the induced norm = (, ). We also use w, v e = e w vdl for the integration along the cell interfaces. To set up the discontinuous Galerkin approximation, we firstly write down the weak formulation of (3.8) ( t u, v) K = d (A i u, xi v) K + d ( n i e,k A i )u, v e, (3.9) where n e,k = (n 1 e,k,..., nd e,k ) is the outward unit normal vector to the edge e in K. Since A i are symmetric, for each e there is an orthogonal matrix S e such that Λ e = diag(λ 1 e,..., λ m e ) = S e ( d ni e,k A i )S T e is diagonal. The numerical scheme can be obtained by applying the upwind flux for each eigen-component, namely ( t u h, v h ) K = e K d (A i u h, xi v h ) K + Λ e ŜS e u h, S e v h e, (3.10) 22 e K

23 where ŜS e u h = ((ŜS e u h ) 1,..., (ŜS e u h ) m ) T and { (ŜS e u h ) k (S e u int h = )k, λ k e < 0, (S e u ext h )k, λ k e > 0. (3.11) Here, u int h (x, t) = lim u y x,y K h (x, t) and u ext (x, t) = lim h u y x,y K c h (x, t). When λ k e = 0, (ŜS e u h ) k will not contribute to the integration, hence one can avoid defining it in this case. Also note that d ni e,k A i = d ni e,k A i for K K = e, and the same S e should be used on both sides. Then ŜS e u h defined in K and K are the same, and the numerical flux is single-valued on the cell interfaces. Furthermore, for the scalar case, (3.11) coincides with the previous definition in Section As before, we define H(w h, v h ) = K K H K (w h, v h ) = K K ( d (A i w h, xi v h ) K ) Λ e ŜS e w h, S e v h e. (3.12) e K H has the following properties, and its proof can be found in Appendix B. Lemma 3.2. v h, w h V, H(v h, v h ) 0 and H(w h, v h ) Ch 1 w h v h, where C depends on A i and the constant in the inverse estimate. With the definition of H, the scheme (3.10) can be written as t u h = L h u h, (L h u h, v h ) = H(u h, v h ), v h V. (3.13) Similar to those in (3.6) and (3.7), Lemma 3.2 implies that L h is semi-negative and L h Ch 1. Hence, one can use Theorem 2.2 to obtain the following corollary. Corollary 3.4. Consider the discontinuous Galerkin approximation of (3.8) with the fourthorder RK time discretization, u n+1 h = P 4 (τl h )u n h, L h defined in (3.11), (3.12) and (3.13). The fully-discrete scheme is strongly stable in two steps, u n+2 h u n h, 23

24 under the CFL condition τ Ch. Here C depends on A i and the constant in the inverse estimate. Remark 3.1. The stability extends to symmetrizable hyperbolic systems with constants coefficients. More specifically, if there is a symmetric positive-definite matrix H such that HA i are symmetric, then under an appropriate CFL condition τ Ch 1, one has u n+2 h H u n h H, where v H = K K K vt Hvdx. Remark 3.2. For symmetric hyperbolic systems with variable coefficients, one can follow similar lines to show L h is semi-bounded, and prove the stability of the fully discretized scheme. But the extension to symmetrizable hyperbolic systems with variable coefficients is non-trivial. 4 Concluding remarks We analyze the stability of the fourth order RK method for integrating method of lines schemes solving the well-posed linear PDE system t u = L(x, t, x )u. The issue of strong stability is of special interests. We consider the ODE system d dt u N = L N u N, with L N +L N 0 and L N independent of time. When L N is normal, the strong stability of the fourth order RK approximation has already been justified by the scalar eigenvalue analysis. But for non-normal L N, we provide a counter example to show that the strong stability can not be preserved whatever small time step we choose. However, the strong stability can actually be obtained in two steps. We prove u n+2 N un N under the time step constraint τ L N c 0 for some constant c 0. This can also be interpreted as the strong stability of the eightstage fourth order RK method composed by two steps of the four-stage method. Then, based on this fact, we extend the stability results to general semi-bounded linear systems after using a perturbation analysis and a froze-coefficient argument. Finally, we apply the results to justify the stability of the fully discretized schemes combining the fourth order RK method and different spatial discretizations, including the spectral Galerkin method and the discontinuous Galerkin method. The corresponding CFL time step restrictions are obtained. 24

25 A Proof of Proposition 1.1 Proof. (1) To prove L N + L T N L N + L T N = = 2 1 ( ) Hence for any u = (u 1, u 2, u 3 ) T R 3, (2) To prove P 4 (τl N ) > 1. By definition, P 4 (τl N ) = u T (L N + L T N)u = 2(u 1 + u 2 + u 3 ) τ + τ2 τ3 + τ4 2τ + 2τ 2 τ 3 + τ4 2τ + 4τ 2 3τ 3 + 4τ τ + τ2 τ3 + τ4 2τ + 2τ 2 τ 3 + τ τ + τ2 2 τ3 6 + τ4 24 Since P 4 (τl N ) = λ max (P 4 (τl N ) T P 4 (τl N )) is the square root of the largest eigenvalue of P 4 (τl N ) T P 4 (τl N ), we define f(λ, τ) = det(λi P 4 (τl N ) T P 4 (τl N )). It suffices to show, for any sufficiently small τ, f(, τ) has a root larger than 1. where After direct calculation, one can obtain One can see f(λ, τ) = λ 3 + c 2 λ 2 + c 1 λ + c 0, ( c 0 = 1 τ τ2 1 6 τ3 + 1 ) 6 24 τ4, c 1 = 3 12τ + 36τ 2 72τ τ 4 211τ τ τ τ τ τ τ τ τ τ τ τ , c 2 = 3 + 6τ 18τ τ 3 46τ τ τ τ τ f(1, τ) = 8τ τ10 3 τ τ τ τ τ τ τ τ τ τ τ τ τ τ , 25.

26 f(2, τ) = 1 + 6τ 18τ τ 3 38τ τ τ6 289τ τ τ τ τ τ19 Hence, for any sufficiently small τ, 383τ τ τ τ τ τ τ τ τ τ τ τ f(1, τ) = 8τ O(τ9 ) < 0, f(2, τ) = 1 + 6τ + O(τ 2 ) > 1. Note f(λ, τ) is a polynomial with respect to λ and τ, hence it is continuous. By the intermediate value theorem, there exists λ(τ) (1, 2) such that f(λ(τ), τ) = 0. Therefore P 4 (τl N ) λ(τ) > 1. Remark A.1. We also provide the numerical examination of Proposition 1.1. The values of P 4 (τl N ) 1 and P 4 (τl N ) 2 1 with a decreasing sequence of τ are listed in Table A.1. As we can see, for this specific case, the one-step strong stability does fail, but the two-step strong stability holds. Table A.1: The numerical examination of Proposition 1.1 and Theorem 2.2 τ P 4 (τl N ) 1 P 4 (τl N ) e e e e e e e e e e e e-12 B Proof of Lemma 3.2 Proof. (1) To prove H(v h, v h ) 0. We denote by v e,h = S e v h and v e,h = ŜS e v h. Since A i are symmetric. Integrating by parts, one has d (A i v h, xi v h ) K = 1 2 e K d n i e,k A i v h, v h e = 1 2 Λ e v e,h, v e,h e. (B.14) e K 26

27 Substitute (B.14) into (3.12), then we obtain H(v h, v h ) = K K Λ e ( 1 2 v e,h v e,h ), v e,h e. e K Each cell interface e is shared by two elements. We sit in either side and call v e,h defined in this element to be v int e,h. v e,h defined from the other side is referred as v ext e,h. Then the summation can be rewritten as H(v h, v h ) = e E = e E ( Λ e ( 1 2 vint e,h v e,h ), v int e,h e + Λ e ( 1 ) 2 vext e,h v e,h ), v ext e,h e ( 1 2 Λ e v int e,h, v int e,h e 1 2 Λ e v ext e,h, v ext e,h e + Λ e v e,h, v ext e,h v int e,h e ). By checking the definition of v e,h, one can prove that H(v h, v h ) = 1 abs(λ e )(v ext 2 e E e,h vint e,h ), (vext e,h vint e,h ) e, where abs(λ e ) = diag( λ 1 e,..., λm e ). Hence H(v h, v h ) is non-negative. (2) To prove H(w h, v h ) Ch 1 w h v h. We will use the notation for different meanings. For scalars, a is the absolute value of a; for vectors, a stands for the Euclidean norm of a; and for matrices, A is the operator norm of A. We also use the notation K = (, ) K. By using the Cauchy- Schwarz inequality and the inverse estimate, one has d i w h, xi v h ) K K K (A K K ( ( d A i w h K xi v h K Ch 1 Ch 1 ) d A i w h K v h K K K ) d A i w h v h. (B.15) And K K e K Λ e ŜS e w h, S e v h e sup k,e λ k e K K ŜS e w h 2 dl e e K S e v h 2 dl. e 27

28 Note that S e are orthogonal, S e v h = v h. By using the inverse inequality, we obtain e S e v h 2 dl = v h 2 dl Ch 1 2 vh K, e and e ŜS e w h 2 dl e w int h 2 + w ext h 2 dl Ch 1 2 wh N(K), where N(K) is the union of K and its neighboring elements. Hence K K e K Λ e ŜS e w h, S e v h e Ch 1 sup λ k e w h N(K) v h K k,e K K Ch 1 sup λ k e w h 2 N(K) k,e K K Ch 1 sup λ k e w h v h. k,e K K v h 2 K (B.16) Combining (B.15) and (B.16), we get H(w h, v h ) Ch 1 w h v h. References [1] B. Cockburn, S. Hou, and C.-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Mathematics of Computation, 54: , [2] B. Cockburn, S.-Y. Lin, and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. Journal of Computational Physics, 84:90 113, [3] B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Mathematics of computation, 52: ,

29 [4] B. Cockburn and C.-W. Shu. The Runge-Kutta local projection P 1 -discontinuous- Galerkin finite element method for scalar conservation laws. RAIRO-Modélisation Mathématique et Analyse Numérique, 25: , [5] B. Cockburn and C.-W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. Journal of Computational Physics, 141: , [6] S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Review, 43:89 112, [7] J.S. Hesthaven, S. Gottlieb, and D. Gottlieb. Spectral methods for time-dependent problems. Cambridge University Press, Cambridge, [8] A. Iserles. A first course in the numerical analysis of differential equations. Cambridge University Press, Cambridge, [9] J.F.B.M. Kraaijevanger. Contractivity of Runge-Kutta methods. BIT Numerical Mathematics, 31: , [10] D. Levy and E. Tadmor. From semidiscrete to fully discrete: stability of Runge-Kutta schemes by the energy method. SIAM review, 40:40 73, [11] J. Shen, T. Tang, and L.-L. Wang. Spectral methods: algorithms, analysis and applications. Springer, Heidelberg, [12] M.N. Spijker. Contractivity in the numerical solution of initial value problems. Numerische Mathematik, 42: , [13] Z. Sun and C.-W. Shu. Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis, to appear. 29

30 [14] E. Tadmor. From semidiscrete to fully discrete: stability of Runge-Kutta schemes by the energy method II. Collected Lectures on the Preservation of Stability under Discretization, Lecture Notes from Colorado State University Conference, Fort Collins, CO, 2001 (D. Estep and S. Tavener, eds.), Proceedings in Applied Mathematics, SIAM, 109:25 49, [15] G. Wanner and E. Hairer. Solving ordinary differential equations II. Springer-Verlag, Berlin,