Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations

Transcription

1 Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations Marlis Hochbruck Tomislav Pažur CRC Preprint 2016/9 March 2016 KARLSRUHE INSTITUTE OF TECHNOLOGY KIT The Research University in the Helmholtz Association wwwkitedu

2 Participating universities Funded by ISSN

3 Numerische Mathematik manuscript No (will be inserted by the editor Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations Marlis Hochbruck Tomislav Pažur January 2015 revised February 2016 Abstract In this paper we study the convergence of the semi-implicit and the implicit Euler methods for the time integration of abstract quasilinear hyperbolic evolution equations The analytical framework considered here includes certain quasilinear Maxwell s and wave equations as special cases Our analysis shows that the Euler approximations are well-posed and convergent of order one The techniques will be the basis for the future investigation of higher order time integration methods and full discretizations of certain quasilinear hyperbolic problems Keywords hyperbolic evolution equations quasilinear Maxwell s equations quasilinear wave equation implicit Euler method semi-implicit Euler method well-posedness error analysis time integration Mathematics Subject Classification (2000 Primary: 65M12 65J15 Secondary: 35Q61 35L90 1 Introduction We consider the time discretization of a quasilinear hyperbolic evolution equation of the form Λ(u(t t u(t = Au(t + Q(u(tu(t u(0 = u 0 (1a on a Hilbert space by two variants of the implicit Euler method Here A is a linear skew-adjoint operator Λ is a symmetric positive definite operator M Hochbruck and T Pažur Institute for Applied and Numerical Mathematics Karlsruhe Institute of Technology Karlsruhe Germany marlishochbruck@kitedu tomislavpazur@kitedu T Pažur AVL AST doo Croatia Zagreb Croatia tomislavpazur@avlcom Corresponding author: M Hochbruck Phone: Fax:

4 2 Marlis Hochbruck Tomislav Pažur on some neighborhood of zero and Q is a nice operator The motivation to consider (1 is that Maxwell s equations with certain quasilinear constitution laws (such as Kerr nonlinearities and quasilinear wave equations fit into this framework The aim of the present paper is to prove well-posedness and convergence of the semi-implicit and the implicit Euler methods applied to (1 In the following we write (1a in the equivalent short form as t u(t = A u(t u(t u(0 = u 0 (1b where A ϕ := Λ(ϕ 1 (A + Q(ϕ Then the approximation of the semi-implicit Euler method is given by (1c u n+1 = u n + τa un u n+1 n = 0 N 1 (2 while the implicit Euler method yields u n+1 = u n + τa un+1 u n+1 n = 0 N 1 (3 Here u n u(t n is an approximation of the exact solution at time t n = nτ where τ > 0 denotes a fixed stepsize Well-posedness of (1 was proved by Kato in [111213] He linearized the problem and used a contraction mapping argument to prove existence and uniqueness of the solution under the assumption that the initial data is smooth enough Using similar techniques Müller [16] refined Kato s result by proving the result for somewhat relaxed assumptions on the initial data Since these refined results are essential for our convergence analysis we use the analytical framework provided in [16] Surprisingly there are only very few convergence results for time integration methods for quasilinear hyperbolic problems The implicit Euler method for nonlinear evolution equations of the form t u(t = N(u has been considered in [101420] for various types of nonlinear operators N: dissipative operators in [20] ω-quasi dissipative operators in [14] and directed L-dissipative operators in [10] These papers show convergence of order 1/2 under the assumption that the numerical solution exists In [4] Crandall and Souganidis showed that the approximations of the semi-implicit Euler method for (1 are well-posed and converge with order 1/2 This enabled them to prove a well-posedness result being equivalent to that of Kato [12] For nonlinear parabolic problems the situation is different Convergence of implicit time integration methods (Runge Kutta and multistep methods for quasilinear problems are given in [ ] and for fully nonlinear parabolic problems in [1 7 18] for instance However since the analytic framework does not fit to hyperbolic problems of the form (1 they cannot be applied here The aim of this paper is to prove convergence of order one for the semiimplicit and the implicit Euler method for the abstract evolution equation (1 in the framework of Kato [12] and Müller [16] We believe that the analysis

5 Implicit Euler methods for quasilinear hyperbolic evolution equations 3 presented here constitutes an important step to prove well-posedness and convergence for a class of higher-order implicit Runge Kutta methods and also to study full discretizations based on finite elements or discontinuous Galerkin discretizations in space In particular Gauss collocation methods would be very interesting to study since their geometric properties are more favorable for hyperbolic problems than Radau methods to which the implicit Euler method belongs However we are aware of the fact that the analysis of higherorder Runge Kutta methods is a nontrivial task It will take significantly more effort to prove well-posedness and stability and to correctly handle the boundary conditions than for the Euler method Nevertheless we are convinced that the techniques presented here for the Euler scheme provide an excellent basis For linear Maxwell s equations such an analysis was carried out in [8] where we proved error bounds for the full discretization with algebraically stable implicit Runge Kutta methods in time and discontinuous Galerkin methods in space Moreover in [9] we showed how the analysis of fully implicit schemes can be used to study the convergence of locally implicit methods which are very efficient for problems whose spatial discretizations is done on locally refined grids The paper is organized as follows In Section 2 we review the analytical framework [12 16] ie we state the precise assumptions on the operators Λ A and Q and give the local well-posedness result Moreover we show that quasilinear Maxwell and wave equations fit into this framework Section 3 contains additional analytical results that are required for the error analysis of the numerical methods Here stability estimates for the resolvents are of great importance For the sake of readability some technical details are postponed to the appendix The semi-implicit Euler method is analyzed in Section 4 Under the same regularity assumptions as in the continuous case the well-posedness and the stability results are proven We derive the error recursion and prove convergence of order one in the L 2 -norm Under additional regularity conditions of the solution we also prove convergence in a stronger norm Section 5 deals with the analysis of the implicit Euler method While for the semi-implicit Euler method well-posedness and stability follow easily by using the stability estimates for the resolvents for the implicit Euler method this is much harder Our proof is based on linearization and a fixed point argument Notation For two normed spaces Y we denote the space of bounded linear operators from to Y by L( Y and for A L( Y we have Ax A Y := max Y x 0 x Given R > 0 we denote by B (R B Y (R the closed balls of radius R around 0 in and Y respectively Finally a b means that there exists a constant c > 0 such that a cb

6 4 Marlis Hochbruck Tomislav Pažur 2 Analytical framework and applications In this section we provide the analytical framework state the known wellposedness result for (1 and present two applications 21 Assumptions and well-posedness The refined analytical framework given in [16] uses three Hilbert spaces ( ( (Y ( Y and (Z ( Z with continuous and dense embeddings Z Y In addition Y is an exact interpolation space between Z and We start with the assumptions on the operator A Assumption 21 (Operator A Let A L(Z Y be a skew-adjoint operator in with Y D(A and let α = A Y Z ie Az Y α z Z for all z Z (4 For Λ we impose the following assumption Assumption 22 (Operator Λ There exist a radius R > 0 and a family of linear operators {Λ(y : y B Y (R} on such that for all y ỹ B Y (R the following holds: (a Λ(y L( is self-adjoint and there is a constant ν > 0 such that Λ(y ν 1 I ie (x Λ(yx ν 1 x 2 Hence Λ(y is invertible in with Λ(y 1 ν for all x (5a (5b (b The range Ran(I Λ(y 1 A is dense in (c There is a constant l > 0 such that Λ(y Λ(ỹ l y ỹ Y (d Λ(y 1 L(Y and there is a constant l Y > 0 such that Λ(y 1 Λ(ỹ 1 Y Y l Y y ỹ Y (5c (5d In the following R always refers to the radius from this assumption on Λ For the operator Q we require the following properties Assumption 23 (Operator Q Let r > 0 be arbitrary We assume that there is a constant µ = µ (r such that Q(y µ for all y B Y (R B Z (r (6a Moreover Q(y L(Z Y for all y B Y (R and there is a constant m Y > 0 such that Q(y Q(ỹ Y Z m Y y ỹ Y for all y ỹ B Y (R (6b

7 Implicit Euler methods for quasilinear hyperbolic evolution equations 5 To obtain error bounds in stronger norms we also need the following assumption on Λ Assumption 24 Let r > 0 be arbitrary We assume that there exists a continuous isomorphism S : Z such that for all z B Y (R B Z (r A S z := SA z S 1 = A z + B(z (7a with linear operators B(z L( being uniformly bounded ie B(z β (7b for some constant β > 0 Remark 25 Let λ 0 := Λ(0 Then the operators Λ(y Λ(y 1 and Q(y are uniformly bounded in the corresponding norms ie for all y B Y (R we have Λ(y λ := λ 0 + lr Λ(y 1 Y Y ν Y := Λ(0 1 Y Y + l Y R Q(y Y Z µ Y := Q(0 Y Z + m Y R (8a (8b (8c The first bound follows from (5c the second from (5d and the third from (6b In the following γ > 0 denotes a given parameter which will be determined later The constants k 0 = (νλ 1/2 1 k 1 = k 1 (γ = 1 2 νlγ (9a c 0 = S S 1 Z Z k 0 1 c 1 = c 0 ν Y (α + µ Y (9b ω = νµ ω = ω + k 0 β (9c L = l (α + µ Y + νm L Y = l Y (α + µ Y + ν Y m Y (9d will be used throughout the paper The following well-posedness result was given in [16 Theorem 341] Theorem 26 Let Assumptions be fulfilled and let κ (0 1 and r > 0 be arbitrary By R 0 = R 0 (κ = κ R c 0 r 0 = r 0 (κ r = κ r c 0 (10 we define two radii satisfying R 0 < R and r 0 < r Then the following assertions hold: (a For each u 0 B Y (R 0 B Z (r 0 there exists a time { } ln κ T = T (κ r R min k 1 (γ + ω κ > 0 γ = γ(r = r c 1 rc 0 L Y c 0

8 6 Marlis Hochbruck Tomislav Pažur and a solution of (1 with u( u 0 = u C([0 T ] Z C 1 ([0 T ] Y u(t Y R and u(t Z r (0 t T (11 (b If v C([0 T ] Z C 1 ([0 T ] Y is another solution of (1 with v(t Y R for all t [0 T ] then v coincides with u on the interval [0 min{t T }] 22 Quasilinear Maxwell s equations Let Ω R 3 be either a bounded domain with boundary Ω or the full space R 3 We consider the Maxwell s equations t D(t x = H(t x t [0 T ] x Ω (12a t B(t x = E(t x t [0 T ] x Ω (12b D(t x = 0 t [0 T ] x Ω (12c B(t x = 0 t [0 T ] x Ω (12d with the constitution relations of the form D(t x = E(t x + P (E(t x B(t x = H(t x + M(H(t x (12e Here P M C n (R 3 R 3 (n N is specified for different examples bellow are vector fields with positive definite matrices I + P (0 and I + M (0 respectively For Maxwell s equations on R 3 let n = s + 1 for some s > 3/2 Then for := L 2 (R 3 6 Y := H s (R 3 6 Z := H s+1 (R 3 6 the problem satisfies the assumptions of Theorem 26 cf [16 Theorem 49] and is therefore well-posed On a bounded domain the well-posedness is proven for the case of Dirichlet boundary condition for the E-field see [16 Theorem 46] Suppose that the boundary Ω is C 4 and that n = 5 Then with := L 2 (Ω 6 Y := H 2 (Ω 6 H 1 0 (Ω 6 Z := {u H 4 (Ω 3 H 1 0 (Ω 3 : u H 1 0 (Ω 3 } 2 the assumptions of Theorem 26 are satisfied Example In applications arising in physics for instance the propagation of light through optical materials in photonic crystals the so called Kerr nonlinearity where the polarization and magnetization are given by P (E = χ E 2 E (χ R M = 0 is of interest cf [319] For χ > 0 the operator Λ is globally invertible and we can replace B Y (R by Y in Assumption 22 Therefore (1 is well-posed for all initial data in Y ie R can be chosen arbitrarily large In the case when χ < 0 the operator Λ is locally invertible (in a neighborhood around 0 and the well-posedness theorem still applies

9 Implicit Euler methods for quasilinear hyperbolic evolution equations 7 23 Quasilinear wave equation Let Ω R d (d 3 be a bounded domain with boundary Ω We are interested in solving a quasilinear wave equation with homogeneous Dirichlet boundary conditions of the form tt w(t x + tt (K w(t x = w(t x t [0 T ] x Ω w(t x = 0 t [0 T ] x Ω (13 For the nonlinearity we assume K C 4 (R and 1 + K (0 > 0 Local well-posedness of a strong solution was shown in [5] It can also be obtained from Theorem 26 by reformulating the problem as a first order system of the form (1 where for u = (u 1 u 2 = (w t w ( 1 0 Λ(u = K A = (u 1 ( cf [16 Theorem 345 and Theorem 412] with := H 1 0 (Ω L 2 (Ω Q(u = ( K (u 1 u 2 Y := (H 2 (Ω H 1 0 (Ω H 1 0 (Ω Z := {u H 3 (Ω H 1 0 (Ω : u H 1 0 (Ω} (H 2 (Ω H 1 0 (Ω 3 Additional properties of the operators For the error analysis of the time integration methods additional properties of the operators are required These are collected in this section Lemma 31 Let Assumption 22 be fulfilled For all y ỹ B Y (R we have (a Λ(y 1/2 λ 1/2 (b (x Λ(y 1/2 x ν 1/2 x 2 for all x (c There is a positive constant l such that Λ(y 1/2 Λ(ỹ 1/2 l y ỹ Y Proof (a follows from (8a and (b follows from (5a (c Since the spectrum σ ( Λ(y J = [ν 1 λ ] for y B Y (R there exists an ellipse Γ in the right complex half-plan which encloses σ ( Λ(y By using the Cauchy integral formula for operators and the resolvent identity we obtain (λ A 1 (λ B 1 = (λ A 1 (A B(λ B 1 (14 Λ(y 1/2 Λ(ỹ 1/2

10 8 Marlis Hochbruck Tomislav Pažur = 1 s 1/2( s Λ(y 1( ( 1ds Λ(y Λ(ỹ s Λ(ỹ 2πi Γ Since Λ(y and Λ(ỹ are self-adjoint we have Λ(y 1/2 Λ(ỹ 1/2 1 2π d(γ J 2 Λ(y Λ(ỹ Γ s 1/2 ds where d(γ J denotes the minimal distance between Γ and J The claim then follows from (5c for l = C(Γ ν λ l To show the convergence of numerical methods in the -norm we need the following assumption Assumption 32 Let r > 0 be arbitrary Then there exist constants l > 0 and m = m (r > 0 such that Λ(y 1 Λ(ỹ 1 Y l y ỹ for all y ỹ B Y (R (15a Q(z Q( z Z m z z for all z z B Z (r (15b Remark 33 The previous assumption can be easily verified for both quasilinear Maxwell s equations (12 (on both R 3 and on a bounded domain and quasilinear wave equations (13 For Maxwell s equations we have Q 0 so (15b is obviously true We thus only sketch how (15a can be shown First we observe that Λ(y = ( I + P (y I + M (y 2 for y = ( y1 y 2 Next we use the resolvent identity (14 for A = P (y 1 and B = P (ỹ 1 and also for the M field The boundedness of the resolvents and the Lipschitzcontinuity of P and M imply Λ(y 1 Λ(ỹ 1 L2 (Ω 6 6 y ỹ The claim then follows from ( Λ(y 1 Λ(ỹ 1 u u L (Ω 6 Λ(y 1 Λ(ỹ 1 L2 (Ω 6 6 u Y y ỹ The following assumption is needed for the proof of the Z-norm convergence Assumption 34 Let r > 0 and z z B Z (r be arbitrary Then there exist positive constants l Z = l Z (r m Z = m Z (r and µ Z = µ Z (r such that Λ(z 1 Λ( z 1 Z Z l Z z z Z (16a Q(z Q( z Z Z m Z z z Z Q(z Z Z µ Z (16b (16c

11 Implicit Euler methods for quasilinear hyperbolic evolution equations 9 Remark 35 All properties can be verified for both quasilinear Maxwell and quasilinear wave equation by straightforward (but rather lengthy computations ie by differentiating the expressions and bounding all the terms (16a additionally requires P (5 and M (5 to be Lipschitz continuous for quasilinear Maxwell s equations with Dirichlet boundary conditions and P C ( s +2 and M C ( s +2 for quasilinear Maxwell s equations on unbounded domains (16b additionally requires K (4 to be Lipschitz continuous for quasilinear wave equations Under the last assumption we also have Λ(z 1 Z Z l Z r + Λ(0 1 Z Z =: ν Z for all z B Z (r (17 Lemma 36 Let Assumptions be fulfilled and let r > 0 be arbitrary Moreover let ϕ ψ B Y (R B Z (r Then the operator A ϕ defined in (1c satisfies the following estimates (a For ν Y and µ Y defined in (8 we have (b For L Y defined in (9d we have A ϕ Y Z ν Y (α + µ Y A ϕ A ψ Y Z L Y ϕ ψ Y (18a (18b (c If in addition Assumption 32 is fulfilled for L defined in (9d we have A ϕ A ψ Z L ϕ ψ (d If in addition Assumption 34 holds and if u satisfies for some r A > 0 we have with L Z = l Z (1 + µ Z + ν Z m Z u Z + Au Z r A (A ϕ A ψ u Z L Z r A ϕ ψ Z Proof (a The inequality follows from (8b (4 and (8c (b Here we use (18c (18d (18e A ϕ A ψ Y Z (Λ(ϕ 1 Λ(ψ 1 A Y Z + (Λ(ϕ 1 Λ(ψ 1 Q(ψ Y Z + Λ(ϕ 1 (Q(ϕ Q(ψ Y Z The claim follows by using (5d and (4 for the first term (5d and (8c for the second term and (8b and (6b for the last term (c We use (15a instead of (5d (5b instead of (8b and (15b instead of (6b (d Analogously to (c but we use (16 and (17 instead of (15 and (5b We have (A ϕ A ψ u Z (l Z ( Au Z + µ Z u Z + ν Z m Z u Z ϕ ψ Z The statement then follows from u Z + Au Z r A

12 10 Marlis Hochbruck Tomislav Pažur 31 Stability estimates For N N and r ζ > 0 we define the function space E := E(N r ζ by E(N r ζ := {ϕ = (ϕ 1 ϕ N Z N : ϕ k Y R ϕ k Z r for k = 1 N ϕ k+1 ϕ k Y ζ for k = 1 N 1} (19 The following stability result is similar to the proofs of Lemma 321 Lemma 329 and Theorem 341 in [16] Because of its importance for the well-posedness results for the numerical solution the sketch of the proof can be found in the appendix Lemma 37 Let γ > 0 and ϕ = (ϕ 1 ϕ N E(N r τγ be given Then (I τa ϕk 1 (I τa ϕj 1 k 0 (1 τω (k j+1 e k1(k jτ (I τaϕk 1 (I τa ϕj 1 Y Y c 0 (1 τ ω (k j+1 e k1(k jτ (I τaϕk 1 (I τa ϕj 1 Z Z c 0 (1 τ ω (k j+1 e k1(k jτ for all τω < 1 in the first inequality τ ω < 1 in the second and third inequality and all 1 j < k N The constants used here are given in (9 4 Semi-implicit Euler method In this section we consider the semi-implicit Euler method (2 which can be written in the equivalent form u n+1 = (I τa un 1 u n = (I τa un 1 (I τa u0 1 u 0 (20 41 Well-posedness and stability We first prove that the approximations defined in (20 are well-posed and uniformly bounded Recall that the constants have been defined in (9 Theorem 41 Let Assumptions be satisfied and let κ (0 1 and r > 0 be arbitrary For each u 0 B Y (R 0 B Z (r 0 where the radii R 0 r 0 are defined in (10 there exists a time ln κ T k 1 (γ + 2 ω > 0 where γ = γ(r = r c 1 c 0 such that for all τ τ 0 = 3/(4 ω and (N + 1τ T there is a unique finite sequence (u n N of semi-implicit Euler approximations (20 satisfying u n Y R and u n Z r n = 1 N (21

13 Implicit Euler methods for quasilinear hyperbolic evolution equations 11 Proof We use induction on n to prove (21 and that the bound u k+1 u k Y γτ for all k = 0 n 1 (22 holds For n = 0 the bounds (21 hold by assumption and the condition (22 is empty Suppose that (21 and (22 hold for some n {0 N 1} Then (u 0 u n E(n + 1 r γτ and we can apply Lemma 37 to (20 which gives u n+1 Y c 0 e (k1+2 ωt u 0 Y u n+1 Z c 0 e (k1+2 ωt u 0 Z (23 Here we used the bound (1 ξ 1 e 2ξ for ξ 3 4 (24 for ξ = τ ω This proves (21 Finally by using a resolvent identity we obtain u n+1 u n = ( (I τa un 1 I u n = τa un (I τa un 1 u n By taking the Y -norm and using (18a and (23 we can bound u n+1 u n Y τν Y (α + µ Y u n+1 Z τc 1 e (k1+2 ωt u 0 Z where c 1 was defined in (9b Inserting γ and T completes the proof 42 Error recursion The Taylor expansion of the exact solution of (1 yields u(t n = u(t n+1 τ t u(t n+1 δ n+1 (25a with defect δ n+1 = Hence we have tn+1 t n 2 t u(t(t n tdt (25b u(t n+1 = u(t n + τa u(tn+1u(t n+1 + δ n+1 (26 We define the error as e n = u n û n û n = u(t n To simplify the following presentation we write Λ n = Λ(u n Λn = Λ ( u(t n A n = A un  n = A u(tn By subtracting (26 from (2 we obtain the error equation e n+1 = e n + τ ( A n u n+1 Ân+1û n+1 δn+1 (27 = e n + τa n e n+1 + τ(a n Ânû n+1 + τ(ân Ân+1û n+1 δ n+1

14 12 Marlis Hochbruck Tomislav Pažur 43 Convergence in the -norm Motivated by the energy techniques presented in [15] for parabolic problems and [8] for linear Maxwell s equations we now prove the convergence of the semi-implicit Euler method in the -norm Lemma 42 Let Assumptions and Assumption 32 be fulfilled Let u C([0 T ] Z C 1 ([0 T ] Y be the exact solution of (1 Then for τ sufficiently small the error e n = u n u(t n of the semi-implicit Euler approximation (2 satisfies e N 2 τm for 0 Nτ T where and e n τνλ ( ρ n := û n+1 û n = tn+1 δ n+1 τ 2 + L r ρ n 2 t n u (tdt (28 M = 2ν ( µ L λ r + l λ 1/2 ν Y (α + µ Y r λ (29 Proof Let Λ 1 = 0 By taking the -inner product of (27 with Λ n e n+1 we obtain (Λ 1/2 n e n+1 Λ 1/2 n 1 e n Λ 1/2 n e n+1 =τ(a n e n+1 Λ n e n+1 + τ((a n Ânû n+1 Λ n e n+1 + τ((ân Ân+1û n+1 Λ n e n+1 + ((Λ 1/2 n Λ 1/2 n 1 e n Λ 1/2 n e n+1 (δ n+1 Λ n e n+1 (30 We bound each of the terms on the right-hand side separately For the first term we use that Λ is self-adjoint A is skew-adjoint and (6a to get τ(a n e n+1 Λ n e n+1 = τ((a + Q(u n e n+1 e n+1 τµ e n+1 2 (31 By applying the Cauchy-Schwarz inequality (18c (8a and (11 for the second and the third term we obtain τ((a n Ânû n+1 Λ n e n+1 τ L λ û n+1 Z e n e n+1 (32 τ ( 2 L λ r e n 2 + e n+1 2 and with ρ n defined in (28 τ((ân Ân+1û n+1 Λ n e n+1 τl λ û n+1 Z û n+1 û n e n+1 τ ( 2 L λ r ρ n 2 + e n+1 2 (33

15 Implicit Euler methods for quasilinear hyperbolic evolution equations 13 To bound the fourth term we use Lemma 31 to get ((Λ 1/2 n Λ 1/2 n 1 e nλ 1/2 n e n+1 l λ 1/2 where we used (21 and τ 2 l λ 1/2 ν Y (α + µ Y r u n u n 1 Y e n e n+1 ( e n 2 + e n+1 2 u n u n 1 Y = τ A n 1 u n Y τν Y (α + µ Y u n Z (34 for the second inequality The latter bound follows from (2 and (18a For the fifth term it holds ( (δ n+1 Λ n e n+1 τ 2 2 λ δ n+1 τ + e n+1 2 (35 The claim now follows by summing (30 for n = 0 N 1 and using the estimates (31 (35 to bound the right-hand side For the left-hand side we use e 0 = 0 to show (Λ 1/2 n e n+1 Λ 1/2 n 1 e n Λ 1/2 n e n where the first inequality follows from Λ 1/2 e N ν 1 e N 2 (36 (v n v n 1 v n = 1 2 v v 2 (v N 2 v v N v N 2 2 (v N 3 v N v N v 0 2 (v 1 v v v ( v 2 v 1 2 for v n = Λ 1/2 n e n+1 (where v 1 = 0 The last inequality in (36 is a consequence of (5a Theorem 43 Let the assumptions of Lemma 42 be fulfilled We additionally assume u L 2 (0 T ; Then for τ sufficiently small the error e n = u n u(t n of the semi-implicit Euler method is bounded by ( T T e N 2 e2mt νλ τ 2 u (t 2 dt + L r u (t 2 dt 0 0 with M = M(r given in (29 ie the method is convergent of order one

16 14 Marlis Hochbruck Tomislav Pažur Proof By a discrete Gronwall inequality and the previous lemma it follows that ( 2 e N 2 e2mt νλ τ δ n+1 τ + L r ρ n 2 for τ 3 4 M 1 The bounds δ n+1 τ 2 τ T 0 u (t 2 dt T ρ n 2 τ u (t 2 dt for the defects δ n+1 and ρ n defined in (25 and (28 respectively imply the stated result 0 44 Convergence in the Z-norm Lemma 44 Let Assumptions and Assumption 34 be fulfilled Let u C([0 T ] Z C 1 ([0 T ] Y be the exact solution of (1 and assume that it satisfies (18d uniformly in t Then for τ sufficiently small the error e n = u n u(t n of the semi-implicit Euler approximation (2 satisfies e N 2 Z τm Z e n+1 2 Z ( + τνκ(s 2 λ δ n+1 τ 2 Z + L Z r A ρ n 2 Z where M Z = M Z (r is given in M Z = 2νκ(S 2( µ + λ β L Zλ r A + l λ 1/2 ν Y (α + µ Y r λ (37 and where κ(s := S Z S 1 Z denotes the condition number of the operator S Proof We multiply (27 by S and use (7a to obtain Se n+1 Se n = τ(a n + B(u n Se n+1 + τs(a n Ânû n+1 + τs(ân Ân+1û n+1 Sδ n+1 By taking the -inner product with Λ n Se n+1 we have (Λ 1/2 n Se n+1 Λ 1/2 n 1 Se n Λ 1/2 n Se n+1 = τ((a n + B(u n Se n+1 Λ n Se n+1 + τ(s(a n Ânû n+1 Λ n Se n+1 + τ(s(ân Ân+1û n+1 Λ n Se n+1 + ((Λ 1/2 n Λ 1/2 n 1 Se n Λ 1/2 n Se n+1

17 Implicit Euler methods for quasilinear hyperbolic evolution equations 15 (Sδ n+1 Λ n Se n+1 We bound each of the terms on the right-hand side separately For the first term we again use that A is skew-adjoint Λ is self-adjoint (6a (8a and (7b to show τ((a n + B(u n Se n+1 Λ n Se n+1 τ(µ + λ β S 2 Z e n+1 2 Z We apply the Cauchy-Schwarz inequality and (18e to bound the second term by τ(s(a n Ânû n+1 Λ n Se n+1 τ S Z (An Ânû n+1 Z Λ n Se n+1 τ 2 S 2 Z λ L Z r A ( e n 2 Z + e n+1 2 Z Similarly as in (33 for the third term it holds τ(s(ân Ân+1û n+1 Λ n Se n+1 τ 2 S 2 Z λ L Z r A ( ρ n 2 Z + e n+1 2 Z where ρ n was defined in (28 We bound the fourth and the fifth term analogously as in the proof of Lemma 42 which gives and ((Λ 1/2 n Λ 1/2 n 1 Se n Λ 1/2 n Se n+1 τ 2 S 2 Z l λ 1/2 ν Y (α + µ Y r (Sδ n+1 Λ n Se n+1 τ 2 S 2 Z λ ( Summing these bounds from 0 to N 1 and using ( e n 2 Z + e n+1 2 Z δ n+1 τ (Λ 1/2 n Se n+1 Λ 1/2 n 1 Se n Λ 1/2 n Se n for the left-hand side proves the lemma 2 Z + e n+1 2 Z S 1 2 Z ν 1 e N 2 Z Theorem 45 Let the assumptions of Lemma 44 be fulfilled and in addition assume u L 2 (0 T ; Z and u L 2 (0 T ; Z Then for τ sufficiently small the error of the semi-implicit Euler method is bounded by ( T T e N 2 Z e2m ZT νκ(s 2 λ τ 2 u (t 2 Z dt + L Zr A u (t 2 Z dt 0 with M Z = M Z (r given in (37 ie the method is convergent of order one Proof The proof is analogous to the proof of Theorem 43 0

18 16 Marlis Hochbruck Tomislav Pažur 5 Implicit Euler method Next we consider the (fully implicit Euler method (3 which we write as u n+1 = (I τa un+1 1 u n (38 In contrast to the semi-implicit Euler method where the approximations are defined via a linear problem here a nonlinear problem has to be solved to compute u n+1 from u n This makes the analysis more involved 51 Well-posedness We start with proving that the approximations are well-posed by following ideas from [16 Theorem 341] which are based on Banach s fixed point theorem Theorem 51 Let Assumptions be satisfied and let κ (0 1 and r > 0 be arbitrary For each u 0 B Y (R 0 B Z (r 0 where the radii R 0 r 0 are defined in (10 there exists a time T = T (κ r R min { ln κ k 1 (γ + 2 ω κ rc 0 L Y } > 0 γ = γ(r = r c 1 c 0 such that for all τ τ 0 = 3/(4 ω and (N + 1τ T there is a unique finite sequence (u n N of implicit Euler approximations (38 which satisfy (21 Proof The space E = E(N r τγ defined in (19 equipped with the metric d(ϕ ψ := max ϕ k ψ k Y k=1n ϕ ψ E is a complete metric space For ϕ = (ϕ 1 ϕ N E we define a function Φ u0 : E N by Φ u0 (ϕ k := (I τa ϕk 1 (I τa ϕ1 1 u 0 k = 1 N It is easy to see that a fixed point of Φ u0 is equivalent to is a solution of (38 ie Φ u0 (ϕ = ϕ ϕ k = (I τa ϕk 1 (I τa ϕ1 1 u 0 k = 1 N For the existence of a unique fixed point we have to prove that Φ u0 is a contraction on (E d We first prove that Φ u0 leaves E invariant Analogously to the proof of Theorem 41 we obtain for T given in the theorem Φ u0 (ϕ k Y c 0 e (k1+2 ωt u 0 Y R Φ u0 (ϕ k Z c 0 e (k1+2 ωt u 0 Z r (39 Φ u0 (ϕ k+1 Φ u0 (ϕ k Y τc 1 e (k1+2 ωt u 0 Z γτ where c 0 and c 1 are defined in (9b Thus Φ u0 (ϕ E

19 Implicit Euler methods for quasilinear hyperbolic evolution equations 17 It remains to prove that Φ u0 is a contraction on E ie Writing G ψ kk = I and d(φ u0 (ϕ Φ u0 (ψ κ d(ϕ ψ ϕ ψ E (40 we have G ψ kj := (I τa ψ k 1 (I τa ψj j < k N Φ u0 (ϕ k Φ u0 (ψ k k 1 = G ψ ( kk i (I τaϕk i 1 (I τa 1 ψk i G ϕ k i 10 u 0 i=0 For the factor in the middle we use the resolvent identity (14 to obtain k 1 Φ u0 (ϕ k Φ u0 (ψ k = τ G ψ kk i 1 (A ϕ k i A ψk i G ϕ k i0 u 0 Lemma 37 and (18b yield i=0 Φ u0 (ϕ k Φ u0 (ψ k Y k 1 τ (Aϕk i A ψk i G ϕ Y Y Y Z i=0 G ψ kk i 1 k i0 u 0 Z k 1 τc 2 0(1 τ ω (k+1 e k1(k 1τ L Y u 0 Z ϕ k i ψ k i Y By taking the maximum over all k we finally obtain the bound d ( Φ u0 (ϕ Φ u0 (ψ T c 2 0e (k1+2 ωt L Y u 0 Z d(ϕ ψ i=0 Therefore Φ u0 is a contraction for the T given in the theorem Remark 52 The interval of existence for the implicit Euler method differs from the one in the continuous case (Theorem 26 only by a factor of two in front of ω This factor comes from the factor two in the estimate (24 which we have used in the well-posedness proof of the semi-implicit Euler method and also in the proof for the implicit Euler method Note that we can choose a factor arbitrarily close to one if ξ is sufficiently small Thus the time interval on which the implicit Euler method is well-posed can be extended to the existence interval of the continuous problem in the limit τ 0

20 18 Marlis Hochbruck Tomislav Pažur 52 Error recursion Subtracting (26 from (3 and using the same notation as in Section 4 we have e n+1 = e n + τ(a n+1 u n+1 Ân+1û n+1 δ n+1 or equivalently e n+1 = e n + τa n+1 e n+1 + τ(a n+1 Ân+1û n+1 δ n+1 (41 53 Convergence in the -norm We proceed by proving the convergence of the method in the -norm Lemma 53 Let Assumptions and Assumption 32 be fulfilled Let u C([0 T ] Z C 1 ([0 T ] Y be the exact solution of (1 Then for τ sufficiently small the error e n = u n u(t n of the implicit Euler approximation (3 satisfies where e N 2 τ M e n τνλ M = M(r ( = 2ν µ + L λ r + l λ 1/2 δ n+1 τ 2 ν Y (α + µ Y r λ (42 Proof By taking the -inner product of (41 with Λ n+1 e n+1 we obtain (Λ 1/2 n+1 e n+1 Λ 1/2 n e n Λ 1/2 n+1 e n+1 =τ(a n+1 e n+1 Λ n+1 e n+1 + τ((a n+1 Ân+1û n+1 Λ n+1 e n+1 + ((Λ 1/2 n+1 Λ1/2 n e n Λ 1/2 n+1 e n+1 (δ n+1 Λ n+1 e n+1 We bound all terms on the right-hand side as in the proof of Lemma 42 and obtain (Λ 1/2 n+1 e n+1 Λ 1/2 n e n Λ 1/2 n+1 e n+1 τµ e n τ L λ û n+1 Z e n τ 2 l λ 1/2 ν Y (α + µ Y u n+1 Z ( e n 2 + e n+1 2 ( + τ 2 2 λ δ n+1 τ + e n+1 2 The claim now follows by summing the last inequality from 0 to N 1 and using the same estimate for the left-hand side as in the proof of Lemma 42

21 Implicit Euler methods for quasilinear hyperbolic evolution equations 19 Theorem 54 Let the assumptions of Lemma 53 be fulfilled and in addition assume u L 2 (0 T ; Then for τ sufficiently small the error e n = u n u(t n of the implicit Euler method is bounded by T e N 2 e2 MT νλ τ 2 u (t 2 dt where M = M(r is given in (42 ie the method is convergent of order one Proof Analogously to the proof of Theorem Convergence in the Z-norm Lemma 55 Let Assumptions and Assumption 34 be fulfilled Let u C([0 T ] Z C 1 ([0 T ] Y be the exact solution of (1 and assume that it satisfies (18d uniformly in t Then for τ sufficiently small the error e n = u n u(t n of the implicit Euler approximation (3 satisfies e N 2 Z τ M Z where M Z = M Z (r is defined as e n+1 2 Z + τνκ(s2 λ δ n+1 τ M Z := 2νκ(S 2( µ + λ β + L Z λ r A + l λ 1/2 ν Y (α + µ Y r λ (43 The condition number κ(s was defined in Lemma 44 Proof We multiply (41 with S and use (7a to obtain Se n+1 Se n = τ(a n + B(u n Se n+1 + τs(a n+1 Ân+1û n+1 Sδ n+1 By taking the -inner product with Λ n+1 Se n+1 we have 2 Z (Λ 1/2 n+1 Se n+1 Λ 1/2 n Se n Λ 1/2 n+1 Se n+1 = τ((a n+1 + B(u n+1 Se n+1 Λ n+1 Se n+1 + τ(s(a n+1 Ân+1û n+1 Λ n+1 Se n+1 + ((Λ 1/2 n+1 Λ1/2 n Se n Λ 1/2 n+1 Se n+1 (Sδ n+1 Λ n+1 Se n+1 (44 The claim now follows by using the same estimates as in the proof of Lemma 44

22 20 Marlis Hochbruck Tomislav Pažur Theorem 56 Let the assumptions of Lemma 55 be fulfilled and assume u L 2 (0 T ; Z Then for τ sufficiently small the error e n = u n u(t n of the implicit Euler method is bounded by T e N 2 Z e2 M Z T νκ(s 2 λ τ 2 u (t 2 Z dt with M Z = M Z (r given in (43 ie the method is convergent of order one Proof The proof is analogous to the proof of Theorem 43 0 A Stability estimates In this appendix we sketch the proof of Lemma 37 For ϕ B Y (R we define inner product (x y ϕ = (Λ(ϕx y With ϕ we denote the space endowed with this inner product From (5a and (8a follows that the associated norm is uniformly equivalent to the -norm ie By using (5c and (45 for ϕ ψ B Y (R we have It follows that λ 1 x 2 ϕ x 2 ν x 2 ϕ x (45 x 2 ϕ = (Λ(ϕx x = (Λ(ψx x + ((Λ(ϕ Λ(ψx x x 2 ψ + l ϕ ψ Y x 2 (1 + lν ϕ ψ Y x 2 ψ x ϕ e k 1τ x ψ for ϕ ψ Y γτ (46 where k 1 := k 1 (γ is defined in (9a For a Banach space V and real numbers C 1 and a > 0 we denote by G(V C a the set of all infinitesimal generators of C 0 -semigroups of type (C a on V We show that for ϕ B Y (R there holds A ϕ G( ϕ 1 ω (47a where ω is defined in (9c which then implies the following bound for the resolvent (I τaϕ 1 ϕ ϕ (1 τω 1 for τω < 1 (47b From (Λ(ϕ 1 Ax x ϕ = 0 Assumption 22(b the fact that A is a closed operator in (since it is skew-adjoint and the norm equivalence (45 we can conclude by using the Lumer-Phillips theorem cf [6 Theorem II315] that Λ(ϕ 1 A generates a contraction semigroup on ϕ Further on Λ(ϕ 1 Q(ϕ ωi is a bounded operator on ϕ and ((Λ(ϕ 1 Q(ϕ ωix x ϕ µ x 2 ω x 2 ϕ (µ ν ω x 2 ϕ = 0 ie Λ(ϕ 1 Q(ϕ ωi is dissipative in ( ϕ Therefore by the perturbation result [6 Theorem III27] we have that A ϕ ωi generates a contraction semigroup on ϕ ie A ϕ ωi G( ϕ 1 0 (47 now follows by the bounded perturbation theorem (cf [6 Theorem III13] We proceed as follows by using (45 (46 and (47 to obtain the -norm estimate For τω < 1 there holds (I τaϕk 1 (I τa ϕj 1 u

23 Implicit Euler methods for quasilinear hyperbolic evolution equations 21 ν 1/2 (I τaϕk 1 (I τa ϕj 1 u ϕk ν 1/2 (1 τω 1 (I τaϕk 1 1 (I τa ϕj 1 u ϕk ν 1/2 (1 τω 1 e k 1τ (I τa ϕk 1 1 (I τa ϕj 1 u ϕk 1 ν 1/2 (1 τω (k j+1 e k 1(k jτ u ϕj k 0 (1 τω (k j+1 e k 1(k jτ u To get the Z-norm estimate we use the operator A S ϕ = A ϕ + B(ϕ defined in (7a For ϕ B Y (R B Z (r by (7b and (45 we obtain B(ϕx ϕ λ 1/2 B(ϕx λ1/2 β x k 0β x ϕ ie B(ϕ ϕ ϕ k 0 β Applying the bounded perturbation theorem again gives that for ϕ B Y (R B Z (r it holds A S ϕ G( ϕ 1 ω where ω is defined in (9c For τ ω < 1 we can write (I τaϕk 1 (I τa ϕj 1 u Z = S 1 (I τa S ϕ k 1 (I τa S ϕ j 1 Su Z S 1 (I Z τa S ϕk 1 (I τa S ϕ j 1 Su and proceed as above This yields the bound in the Z-norm Since Y is an exact interpolation space between Z and the second inequality follows immediately Acknowledgements The authors thank Roland Schnaubelt and Dominik Müller for helpful discussions on the well-posedness of the Euler approximations We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG through RTG 1294 and CRC 1173 References 1 G Akrivis and M Crouzeix Linearly implicit methods for nonlinear parabolic equations Math Comp 73(246: (electronic G Akrivis M Crouzeix and C Makridakis Implicit-explicit multistep methods for quasilinear parabolic equations Numer Math 82(4: K Busch G von Freymann S Linden S F Mingaleev L Tkeshelashvili and M Wegener Periodic nanostructures for photonics Phys Reports 444(3-6: M G Crandall and P E Souganidis Convergence of difference approximations of quasilinear evolution equations Nonlinear Anal 10(5: W Dörfler H Gerner and R Schnaubelt Local wellposedness of a quasilinear wave equation J Appl Ana 2016 to appear 6 K-J Engel and R Nagel One-parameter semigroups for linear evolution equations volume 194 of Graduate Texts in Mathematics Springer-Verlag New York 2000 With contributions by S Brendle M Campiti T Hahn G Metafune G Nickel D Pallara C Perazzoli A Rhandi S Romanelli and R Schnaubelt 7 C González A Ostermann C Palencia and M Thalhammer Backward Euler discretization of fully nonlinear parabolic problems Math Comp 71(237: (electronic 2002

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