Runge–Kutta time semidiscretizations of semilinear PDEs with non-smooth data

We study semilinear evolution equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\mathrm dU}{\mathrm dt}=AU+B(U)$$\end{document}dUdt=AU+B(U) posed on a Hilbert space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal Y$$\end{document}Y, where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal Y_\ell = D(A^\ell )$$\end{document}Yℓ=D(Aℓ) to itself, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in I \subseteq [0,L]$$\end{document}ℓ∈I⊆[0,L], \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L \ge 0$$\end{document}L≥0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0,L \in I$$\end{document}0,L∈I. In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge–Kutta method in time, retaining continuous space, and prove convergence of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^{p\ell /(p+1)})$$\end{document}O(hpℓ/(p+1)) for non-smooth initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U^0\in \mathcal Y_\ell $$\end{document}U0∈Yℓ, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \le p+1$$\end{document}ℓ≤p+1, for a method of classical order p, extending a result by Brenner and Thomée for linear systems. Our approach is to project the semiflow and numerical method to spectral Galerkin approximations, and to balance the projection error with the error of the time discretization of the projected system. Numerical experiments suggest that our estimates are sharp.


Introduction
We study the convergence of a class of A-stable Runge-Kutta time semidiscretizations of the semilinear evolution equation dU dt = AU + B(U ) (1.1) with arbitrary ∈ [0, p + 1] and also provides sufficient conditions for when q = q( ) = p /(p + 1) for all ∈ [0, p + 1] (which are satisfied in our setting).
For splitting methods, where the linear part of the evolution equation is evaluated exactly, a higher order of convergence has been obtained for specific choices of and specific evolution equations in [13] and [6], see also Example 5.4 below.While splitting methods are very effective for simulating evolution equations for which the linear evolution e tA can easily be computed explicitly, Runge-Kutta methods are still a good choice when an eigen-decomposition of A is not available, for example for the semilinear wave equation in an inhomogeneous medium, see Example 2.7.Moreover, the simplest example of a Gauss-Legendre Runge-Kutta method, the implicit mid point rule, appears to have some advantage over split step time-semidiscretizations for the computation of wave trains for nonlinear Schrödinger equations because the latter introduce an artificial instability [18].
For Runge-Kutta time semidiscretizations of dissipative evolution equations, where A is sectorial, a better order of convergence can be obtained, see [10] for the linear case and [11,12] and references therein for the semilinear case.Note that our approach is different from the approach of [11,12].In [11,12] some smoothness of the continuous solution is assumed and from that a (fractional) order of convergence is obtained, using the variation of constants formula.The order of convergence obtained in [11,12] is in general lower than in the linear case (where full order of convergence is obtained in the parabolic case [10]), but no extra assumptions on the nonlinearity B(U ) of the PDE are made.In particular in [12,Theorems 4.1 and 4.2] the existence of (p s + 2) time derivatives of the continuous solution U (t) of a semilinear parabolic PDE (1.1) is assumed, where p s is the stage order of the method.This assumption is then used to estimate the error of the numerical approximation of the inhomogenous part of the variation of constants formula.Here the stage order p s comes into play.Note that if the nonlinearity B(U ) of the evolution equation (1.1)only satisfies the standard assumption rather than our assumption (B), i.e., is smooth on Y only (so that the Hilbert space scale is trivial with L = 0) then the existence of U (t) can be guaranteed for U 0 ∈ Y 1 by semigroup theory [17], but it is not clear whether higher order time derivatives of the solution U (t) of (1.1) exist as assumed in [12] -therefore in [12] also timedependent perturbations of (1.1) are considered.In this paper we instead take the approach of making assumptions (namely condition (B) on the nonlinearity B(U ) of the evolution equation and the condition that U 0 ∈ Y ) which are straightforward to check and guarantee the existence of the time derivatives of the continuous solution U (t) up to order k ≤ .We then obtain an order of convergence O(h p /(p+1) ) of the Runge-Kutta discretization which is identical to the order of convergence in the linear case [3,9].In [11,Theorem 2.1] some smoothness of the inhomogeneity of the PDE is obtained from the smoothing properties of parabolic PDEs, and this is used to prove an order of convergence h log h, without the assumption of the existence of higher time derivatives of the continuous solution U (t).Here we do not consider parabolic PDEs, so that we cannot use this strategy.
Alonso-Mallo and Palencia [2] study Runge-Kutta time discretizations of inhomogeneous linear evolution equations where the linear part creates a strongly continuous semigroup.Similarly as in [12] they obtain an order of convergence depending on the stage order p s of the Runge-Kutta method.They assume the continuous solution U (t) to be (p + 1)-times differentiable in t, but in their context the condition U (t) ∈ D(A p−ps ), where p is the order of the numerical method, is in general not satisfied due to the inhomogeneous terms in the evolution equation, and this leads to a loss in the order of convergence compared to our results.Note that in our setting, due to our condition (B) on the nonlinearity, provided U (0) ∈ Y p+1 we have U (t) ∈ D(A p+1 ) = Y p+1 and U (t) is p + 1 times differentiable in t (in the Y norm) and so we get full order of convergence in this case (see [15]).Calvo et al [4] study Runge-Kutta quadrature methods for linear evolution equations U (t) = A(t)U (t) which are well-posed and prove full order convergence if the continuous solution U (t) has p + 1 time derivatives; they also obtain fractional orders of convergence as in [3] for solutions U (t) ∈ Y with < p + 1.
We proceed as follows: in Section 2 we introduce the class of semilinear evolution equations that we consider in this paper, give some examples, review existence and regularity results of [17,15] for the semiflow, and adapt them to the case of noninteger .In Section 3 we introduce a class of A-stable Runge-Kutta methods.We review existence and regularity of these methods when applied to the semilinear evolution equation (1.1) and a convergence result for sufficiently smooth initial data from [15].In Section 4 we study the stability of the semiflow and numerical method under spectral Galerkin truncation, and establish estimates for the projection error.Lemma 4.2 and 4.3 are established in [16] for integer values of ; for completeness we review the proofs, which also work for non-integer .In Section 5 we prove our main result on convergence of A-stable Runge-Kutta discretizations of semilinear evolution equations for non-smooth initial data.In Section 6 we generalize our result to nonlinearities B(U ) which are defined on domains other than balls.

Semilinear PDEs on a scale of Hilbert spaces
In this section we introduce a suitable functional setting for the class of equations we subsequently study.We review results from [17,15] on the local well-posedness and regularity of solutions of (1.1) and give examples.
For a Hilbert space X we let be the closed ball of radius R around U 0 in X .We make the following assumptions on the semilinear evolution equation (1.1): (A) A is a normal linear operator on Y that generates a strongly continuous semigroup of linear operators e tA on Y in the sense of [17].
It follows from assumption (A) that there exists ω ∈ R with see [17].In light of (A) we define the continuous scale of Hilbert spaces Y = D(A ), ≥ 0, Y 0 = Y.Thus the parameter is our measure of smoothness of the data.For m > 0 we define P m to be the spectral projection of A to spec(A) ∩ B m C (0), let Q m = id −P m and set P = P 1 , Q = id −P.We endow Y with the inner product which implies We deduce from assumption (A) that for u ∈ Y, lim m→∞ P m u = u, and from (2.2) the estimates Remark 2.1.When lies in a discrete set such as N 0 , for > 0 often the inner product The reason why we do not use this inner product here is that (2.2) is continuous in as → 0, but the graph inner product (2.5) is not: we have To formulate our second assumption, on the nonlinearity B, we introduce the following notation: for Banach spaces X , Z, we denote by E i (X , Z) the space of i-multilinear bounded mappings from X to Z.For U ⊆ X we write C k b (U, Z) to denote the set of k times continuously differentiable functions F : int U → Z such that F and its derivatives D i F are bounded as maps from the interior int U of U to E i (X , Z) and extend continuously to the boundary of int there are examples of continuous functions F : U → Z where U is closed and bounded, which do not lie in C b (U, Z), see e.g.[15,Remark 2.3].In the following for ∈ R let be the largest integer less than or equal to and be the smallest integer greater or equal to .Moreover for R > 0 and ≥ 0 we abbreviate (2.7)We seek a solution U (•) ∈ C([0, T ]; Y ) of (1.1) for some T > 0, ∈ I, with initial data U (0) = U 0 ∈ Y , and write Φ t (U 0 ) ≡ Φ(U 0 , t) ≡ U (t).The following result is an extension of Theorem 2.4 of [15], see also [17], to non-integer and provides well-posedness and regularity of the semiflow Φ t under suitable assumptions.Theorem 2.2 (Regularity of the semiflow).Assume that the semilinear evolution equation (1.1) satisfies (A) and (B).Let R > 0. Then there is T * > 0 such that there exists a semiflow Φ which satisfies with uniform bounds in U ∈ B R/2 .The bounds on T * and Φ depend only on R, ω from (2.1), and the bounds afforded by assumption (B) on balls of radius R.
Proof.The proof of (2.8) is an application of a contraction mapping theorem with parameters to the map The solution W (U, T )(t) = Φ tT (U ) of (1.1) is obtained as a fixed point of (2.9) Y (0) as in [15].Here Π : In order to apply the contraction mapping theorem we first check that Π(W, (2.10) follows from the fact ∈ I − that the above argument applies with Y replaced by Y −j , j = 0, . . ., k. Hence there is some the t derivatives up to order k can then be obtained by implicit differentiation of Π(W (U, T ), U, T ) = W (U, T ) with Π defined above which implies that Φ Note that this theorem extends to mixed (U, t) derivatives which are, however, in general only strongly continuous in t, see [15] for details.For our purposes in this paper the above theorem is sufficient.

.12)
Here P 0 is the spectral projector of Ã to the eigenvalue 0. Since the Laplacian is diagonal in the Fourier representation with eigenvalues −k 2 for k ∈ Z, the eigenvalue problem for A separates into 2 × 2 eigenvalue problems on each Fourier mode, and it is easy to see that the spectrum of A is given by spec A = {ik : k ∈ Z} \ {0}.
Note that P 0 Ã has a Jordan block and is hence included with the nonlinearity B.
We denote the Fourier coefficients of a function u ∈ L (2.14) In the setting of the semilinear wave equation, we have H ) for all R > 0 and therefore that (B) holds, noting that Y is as in (2.15).Here we abbreviated H := H ([0, 2π]; R).
Example 2.5 (Semilinear wave equation, Dirichlet boundary conditions).When endowed with homogeneous Dirichlet boundary conditions u(t, 0) = u(t, π) = 0 the linear part A of the semilinear wave equation (2.11) still generates a unitary group.In this case we have P 0 = 0, A = Ã, and ), where ∆ denotes the Laplacian with Dirichlet boundary conditions.By [8] for If V : R → R is analytic and even so that f = −V satisfies the required boundary conditions, the conclusions of Lemma 2.9 a) apply to f = −V on the spaces sufficient to satisfy either of those two constraints on , at least one of which is always true.So in this example condition (B) is satisfied with I = [0, L] for any L ≥ 0. Moreover the condition that V is even may be relaxed to the requirement that V (2j+1) (0) = 0 for 0 ≤ 2j ≤ L + for / ∈ 3/2 + 2N 0 .If V : R → R is analytic, then the conclusions of Lemma 2.9 a) apply to f = −V on the spaces . This follows from the fact that all terms in the sum obtained from computing ∂ 2j+1 x f (u) contain at least one odd derivative of u of order at most 2j + 1, so that the required boundary conditions for f are satisfied.Hence Condition (B) is satisfied for any L ≥ 0 with I = [0, L].
Example 2.7 (A semilinear wave equation in an inhomogeneous material).Instead of (2.11), let us consider the non-constant coefficient semilinear wave equation with periodic boundary conditions where ) on [0, 2π] with periodic boundary conditions, where V (u, ū) is assumed to be analytic as a function in u 1 = Re (u) and u 2 = Im (u).Setting U = (u 1 , u 2 ), we can write (2.16) in the form (1.1) with The Laplacian is diagonal in the Fourier representation (2.13) with eigenvalues −k 2 and L 2 ([0, 2π]; C)-orthonormal basis of eigenvectors e ±ikx / √ 2π where k ∈ Z.Hence, the spectrum of A is given by spec A = {−ik 2 : k ∈ Z} and A is normal and generates a unitary group on L 2 ([0, 2π]; C) and, more generally, on every H ([0, 2π]; C) with ≥ 0.
By Lemma 2.9 a) below the nonlinearity B(U ) defined in (2.17) is analytic as map from H ([0, 2π]; R 2 ) to itself for every > 1/2.Hence, assumption (B) holds for the nonlinear Schrödinger equation (2.16) When we equip the nonlinear Schrödinger equation (2.16) with Dirichlet (Neumann) boundary conditions we need to require that ) and, for Dirichlet boundary conditions, we need the potential V to be even or satisfy for Neumann boundary conditions.

The nonlinearities of the PDEs in the above examples are superposition operators
or restrictions of such operators to spaces encorporating boundary conditions.To prove that these superposition operators satisfy assumption (B) we have employed the following lemma.Part a) of this lemma has already been stated in slightly different form in [7,14], and parts b) and c) follow from [15].Lemma 2.9 (Superposition operators).Let Ω ⊆ R n be an open set satisfying the cone property.a) Let ρ > 0 and let f : H → H and its derivatives up to order N are bounded with N -dependent bounds for arbitrary

.18)
Here with R-dependent bounds.c) Let D, f and j be as in b) and let L > n/2 be such that L ≤ N .Then with D defined as in (2.18).
Proof.We restrict to the case see, e.g., [1].Let f be analytic on B ρ C and let be the Taylor series of f around 0 for |z| ≤ ρ.Let g : R → R be its majorization By applying the algebra inequality (2.19) to each term of the power series expansion (2.20) of f (u), we see that the series converges for every u ∈ H provided > n/2, and that where c is as in (2.19), R ≤ ρ/c and a 0 = 0 if Ω is unbounded.In other words, f is analytic and bounded as function from a ball of radius R around 0 in H = H (Ω; C) to H . Similarly we see that the same holds for the derivatives of f .To prove b) note that D is well-defined because by the Sobolev embedding theorem H j (Ω; R) ⊆ C b (Ω; R).In [15, Theorem 2.12], the statement was proved in the case n = 1.The extension to the case n > 1 is straightforward.Here let us just illustrate the idea of the proof for the example n = 1, N = 1 and j which is again true by the Sobolev embedding theorem.
To prove c) note that for ∈

Runge-Kutta time semidiscretizations
In this section we apply an A-stable Runge-Kutta method in time to the evolution equation (1.1), and establish well-posedness and regularity of the numerical method on the infinite dimensional space Y.
Given an (s, s) matrix a, and a vector b ∈ R s , we define the corresponding Runge-Kutta method by where Here, W 1 , . . ., W s are the stages of the method, we understand A to act diagonally on the vector W , i.e., (AW ) i = AW i , and We define and re-write (3.1a) as and (3.1b) as where S is the stability function, given by In the following C − 0 = {z ∈ C : Re(z) ≤ 0}.We assume A-stability of the numerical method as follows (cf.[12]): The following result is needed later on, see also [15,Lemmas 3.10,3.11,3.13]:Lemma 3.2.Under assumptions (A), (RK1) and (RK2) there are Finally there are c S,k > 0 with and, with Analogously to Theorem 2.2, we require a well-posedness and regularity result for the stage vectors W i , i = 1, . . ., s, and the numerical method Ψ h .The following result is an extension of [15,Theorem 3.14] to non-integer values of .Theorem 3.3 (Regularity of numerical method).Assume that the semilinear evolution equation (1.1) satisfies (A) and (B), and apply a Runge-Kutta method subject to conditions (RK1) and (RK2).Let R > 0. Then there is h * > 0 such that there exist a stage vector W and numerical method Ψ which satisfy with uniform bounds in U ∈ B r .The bounds on h * , Ψ and W depend only on R, (3.5), those afforded by assumption (B) on balls of radius R and on a, b as specified by the numerical method.
Proof.As in [15] we compute W as fixed point of the map Π : using (3.2).To be able to apply the contraction mapping theorem we need to check that Π ) with N derivatives in U .This proves statements (3.6a) and also (3.6c) in the case k = 0 for W . Due to (3.3), these statements also hold true for Ψ.In the case k = 0 it follows from the that ∈ I − that the above argument also holds on Y −j , j = 0, . . ., k. Hence there is some As shown in [15] for U ∈ B r the h derivatives up to order k can then be obtained by implicit differentiation of Π(W, U, h) = W (U, h) with Π defined above and by differentiating (3.3), cf. the proof of Theorem 2.2.This then implies (3.6c).
A discretization y n+1 = ψ h (y n ) of an ordinary differential equation (ODE) dy dt = f (y) is said to be of classical order p if the local error, i.e., the one-step error, of the numerical method is given by the Taylor remainder of order p + 1, When considering the local error of a semidiscretization of a PDE on a Hilbert space Y, the derivatives of the semiflow and numerical method in time and step size respectively are not necessarily defined on the whole space Y.To obtain global error estimates for semidiscretizations of PDE problems analogous to the familiar results for ODEs, we must consider the local error as a map Z → Y, where Z is a space of higher regularity.Using the regularity results for the semiflow and its discretization in time, Theorems 2.2 and 3.3, the following can be shown (see [15,Theorem 3.20]): if (A), (B), (RK1) and (RK2) hold, and (in our notation) ∈ I − , ≥ p + 1 then for fixed T > 0, R > 0 there exist constants c 1 , c 2 , h * > 0 such that for every solution Φ provided that nh ≤ T .In this paper we study the case where the solution U (t) satisfies U (t) ∈ Y with < p + 1, by means of Galerkin truncation.

Spectral Galerkin truncations
In this section we consider the stability of the semiflow Φ t of (1.1), and the numerical method Ψ h defined by (3.Then there is T * > 0 such that for m ≥ 0 there exists a projected semiflow Φ m which satisfies with uniform bounds in U ∈ B R/2 and m ≥ 0. The bounds on T * and Φ m , depend only on R, ω from (2.1), and those afforded by assumption (B) on balls of radius R.
In the case B ≡ 0 it is clear that for U 0 ∈ Y we have the estimate With the presence of a nonlinear perturbation B = 0 a similar result can be obtained by a Gronwall type argument as shown in the lemma below, which gives an appropriate bound for the error of the semiflow incurred in Galerkin truncation.Note that similar results for mixed higher order derivatives in time and initial value are obtained, for integer in [16, Theorems 2.6 and 2.8].
Here m * and the order constant depend only on δ, R, T , (2.1) and the bounds afforded by (B) on balls of radius R + δ.
Proof.The statement is shown for integer in [16].We review the argument, which also works for arbitrary ∈ I. To prove (4.3b) we use the mild formulation (2.9) for Φ and Φ m .We find where M = M 0 [R + δ] a bound of DB as map from B R+δ 0 to E(Y), see condition (B), and we choose m * > 0 big enough such that Thus, applying a Gronwall type argument, we obtain (4.3b).
We also consider an s-stage Runge-Kutta method applied to the projected semilinear evolution equation (4.1).We denote by w m = w m (u 0 m , h) the stage vector of this map, and by ψ h m (u 0 m ) the one-step numerical method applied to the projected system (4.1) and define W m = w m • P m , Ψ h m = ψ h m • P m .Similar to Lemma 4.1 and Lemma 4.2, we have the following results regarding the existence, regularity and error under truncation for the projected numerical method.Note that similar results have been obtained, for integer , and mixed derivatives in [16, Theorems 3.2 and 3.6].

Lemma 4.3 (Regularity of projected numerical method and projection error).
Assume that the semilinear evolution equation (1.1) satisfies (A) and (B), and apply a Runge-Kutta method subject to conditions (RK1) and (RK2).Let R > 0. Then there is h * > 0 such that for m ≥ 0 there exist a stage vector W m and numerical method Ψ m of the projected system (4.1) which satisfy for i = 1, . . ., s, where r is as in and sup (4.5d) The bounds on h * , Ψ m and W m and the order constants depend only on R, (3.5), those afforded by assumption (B) on balls of radius R and on a, b as specified by the numerical method.
Proof.The statements (4.5a) and (4.5b) are shown exactly as in the proof of Theorem 3.3 and (4.5c), (4.5d) are shown for integer in [16].The same arguments are valid for arbitrary ∈ I as well, we review the proof for completeness.From the formulation (3.2) of the stage vectors with an order constant uniform in U ∈ B r .Here M = M 0 [R] and we used (3.5b) and (2.4).Solving for W (U, h) − W m (U, h) Y s and taking the supremum over h ∈ [0, h * ] and U ∈ B r we get (4.5c).
Similarly for the numerical method using (3.3), (3.5) and (2.4) we estimate Here we used (4.5c) in the last line.

Trajectory error bounds for non-smooth data
In this section we consider the convergence of the global error as h → 0 for non-smooth initial data.As mentioned above, cf.(3.10), [15,Theorem 3.20] states that we have given sufficient regularity of the semiflow and time semidiscretization to bound the local error given by the Taylor expansion to order p + 1 as a map see (3.9).As stated by Theorems 2.2 and 3.3, this is the case provided ∈ I − , ≥ p+1.In this paper we study the order q = q( ) of convergence of the global error for non-smooth initial data U 0 ∈ Y , ∈ I − , < p + 1, such that E n (U, h) = O(h q ) and show that we obtain q( ) = p /( + 1) as Brenner and Thomée [3] and Kovács [9] did for linear strongly continuous semigroups.
The implicit midpoint rule, the simplest Gauss-Legendre method, satisfies the conditions (RK1) and (RK2), see Example 3.1 with p = 2. Figure 1 shows the order of convergence of the implicit midpoint rule applied to the semilinear wave equation (2.11) with V (u) = u − 4u 2 for = j/2, j = 0, . . ., 6, on the integration interval t ∈ [0, 0.5], using a fine spatial mesh (we use N = 1000 grid points on [0, 2π]).As initial values we choose U 0 = (u 0 , v 0 ) ∈ Y where Here c u and c v are such that U 0 Y = 1, with U 0 = (u 0 , v 0 ), and = 10 −8 .From Theorem 2.2, with Y replaced by Y , we know that there is some T * > 0 such that Φ t (U 0 ) ∈ B R for U 0 ∈ Y so that the assumption (5.20) of our convergence result, Theorem 5.3 below, is satisfied.We integrate the semilinear wave equation with the above initial data for the time steps h = 0.1, 0.095, 0.09, 0.085, . . ., 0.05, when > 0. At = 0, to reduce computational effort, we only used the time steps h = 0.1, 0.09, . . .0.05.To estimate the trajectory error, we compare the numerical solution to a solution calculated using a much smaller time step, h = 10 −3 for > 0 and h = 10 −4 for = 0. From the assumption E n (h) = ch q we get log E n (h) = log c + q log h.Fitting a line to those data, we take the gradient of the line as our estimated order of convergence of the trajectory error.The decay in q( ) as decreases from 3 is clearly shown.Note that the order of convergence does not decrease to exactly 0 at = 0 and is slightly better than predicted by our theory when = 2.5.This is because we simulate a space-time discretization rather than a time semidiscretization.Moreover at = 0, despite the fact that we already use a finer time step size, the approximation of the exact solution is not that accurate as the order of convergence for the time-semidiscretization vanishes at = 0.In the rest of this section, equipped with the results of Section 4 on the stability of the semiflow and the numerical method under Galerkin, truncation we estimate the growth with m of the local error of a Runge-Kutta method (3.1), subject to (RK1) and (RK2), applied to the projected equation (4.1) subject to (A) and (B) for non-smooth initial data.In this setting, by coupling m and h and balancing the projection error and trajectory error of the projected system, we obtain an estimate for q( ) that describes the convergence of the numerical method for the semilinear evolution equation (1.1) as observed in Figure 1, see Section 5.2.

5.1.
Preliminaries.We start with some preliminary lemmas.
and for all k ∈ N 0 , k ≤ we have with bounds uniform in U 0 and m ≥ m * .Further, choose k ∈ N 0 with ≤ k ≤ N .Then for all U 0 satisfying (5.3a), (5.3b) still holds, but with m-dependent bounds which are uniform in U 0 .Moreover for all such U 0 , ≤ k ≤ N , with bounds uniform in U 0 .The bounds and order constants only depend on T , R, (2.1) and the bounds from assumption (B).
Proof.Due to Lemma 4.1 statement (5.3b) is non-trivial only if ≥ 1.In this case let u m (t) = Φ t m (U 0 ).From Lemma 4.1 with Y replaced by Y , using (5.3b) we also get u m ∈ C b ([0, T ]; B R ).From (5.3a) and (4.1) we conclude that ) with bounds uniform in m ≥ m * and U 0 satisfying (5.3a).That proves (5.3b) for k = 1.If ≥ 2 then from (4.1) we get for k ≤ with uniform bounds in m ≥ m * and in all U 0 satisfying (5.3a).This proves (5.3b) for k ≤ with m independent bounds.To prove (5.3c) we proceed by induction over k = , . . ., N .We consider the cases < 1 and ≥ 1 separately.If < 1 then from (4.1) we have with order constant independent of m ≥ m * and of U 0 satisfying (5.3a).This then immediately shows (5.3c) for k = = 1.If ≥ 1, ∈ Z then the start of the induction is k = , and the left hand side of (5.3c) is bounded by (5.3b).
If ≥ 1, / ∈ Z then the start of the induction is k = > .Using (5.4) we can bound the -th derivative independent of m in the Y − norm.Using the Faà di Bruno formula [5] we find that for any i ∈ N, i < N , where β = j 1 + • • • + j i and the sum is over all j α ∈ N 0 , α = 1, . . ., i, with We consider (5.5) with i replaced by .Then the second term in the last line of (5.5) is bounded independent of m ≥ m * due to (5.3b).Furthermore, since ∂ t u m ∈ Y − by (5.4) with uniform bound in m ≥ m * , we estimate where we have used the first inequality of (2.4).So (5.3c) also holds true for when > 1, / ∈ Z. Now fix an integer k and assume that (5.3c) holds for all integers i such that ≤ i ≤ k.We now use (5.5) with i = k to estimate ∂ k+1 t u m Y .By the first inequality of (2.4) and the induction hypothesis the first term on the second line of (5.5) is O(m k+1− ).Moreover, by (5.3b) and the induction hypothesis, the Y norm of the second term is of order O(m n ) with n = 0 if j + . . .+ j k = 0 and Thus we see that the right hand term of (5.5), with i = k, is O(m k+1− ) as well.
Lemma 5.2 (m-dependent bounds for derivatives of Ψ m and W m ).Assume that the semilinear evolution equation (1.1) satisfies (A) and (B), and apply a Runge-Kutta method subject to (RK1) and (RK2).Choose ∈ I − and k ∈ N 0 with ≤ k ≤ N .Let R > 0 and define r as in (3.6b).Then there is h * > 0 such that for m ≥ 0 and i = 1, . . ., s, with m-dependent bounds which are uniform in U ∈ B r .Moreover The order constants in (5.7) depend only R, (3.5), a and b from the numerical method and the bounds afforded by (B) on balls of radius R.
for i = 1, . . ., s, j = 1 . . ., , with bounds independent over m ≥ 0 and U ∈ B r .From (3.3) we formally obtain (5.9)By (3.5c) and (2.4) there are h * > 0, c S,k such that for all h ∈ [0, h * ] and k ≥ (5.10) In addition (3.5d) shows that for n ∈ N with n − 1 ≥ Using (5.11) (with replaced by − j and n by k − j) and (5.8), we can estimate the j-th term in the sum of (5.9) for 0 ≤ j ≤ ≤ k as follows: (5.12) To obtain the first estimate of (5.7) assume that there is b j > 0 such that for all h ∈ [0, h * ], U ∈ B r and k ≥ j ≥ .This will be proved below.Then, using (5.11) and (5.13) we can estimate the j-th term in the sum of (5.9) for j ≥ as follows: (5.14) These estimates, with (5.9) and (5.10), then prove the first estimate of (5.7).
To prove (5.13) and the second estimate of (5.7), differentiate (3.2) k times in h: (5.15)By (3.5d) and (2.4), for k ≥ , sup Now we use the Faà di Bruno formula (5.5) again: where β = j 1 + • • • + j k and the sum is over all j α ∈ N 0 , α = 1, . . ., k with We see that all terms on the right hand side of (5.18) contain h-derivatives of order at most k − 1 and are therefore bounded and in particular O(m k− ), except when β = j k = 1 and j α = 0 for α = k.So we obtain sup Substituting this into (5.17)gives the second estimate of (5.7) for k = and h * small enough.Resubstituting this estimate into (5.19) also shows (5.13) for k = .Now assume these estimates hold true for all k ∈ N 0 with ≤ k ≤ k − 1 and let k ≤ N .Then, using the induction hypothesis and the above estimates, in particular (5.12), (5.13), (5.14) and (5.16), all terms in (5.15) are O(m k− ) except when j = k in the sum.We deduce that (5.17) remains valid under the induction hypothesis.Moreover, by the induction hypothesis, each term in the sum of the Faà di Bruno formula (5.18) with j k = 0 is of order O(m n ) with n = 0 if j + . . .+ j k−1 = 0 and Hence (5.19) remains valid, and we deduce (5.13) and the second estimate of (5.7) as before.

5.2.
Trajectory error for nonsmooth data.Now we are ready to prove our main result: Theorem 5.3 (Trajectory error for nonsmooth data).Assume that the semilinear evolution equation (1.1) satisfies (A) and (B) and apply a Runge-Kutta method (3.1) subject to (RK1) and (RK2).Let ∈ I − , 0 < ≤ p + 1, and fix T > 0 and R > 0. Then there exist constants h * > 0, c 1 > 0, c 2 > 0 such that for every and for all h ∈ [0, h * ] we have provided that nh ≤ T .The constants h * , c 1 and c 2 depend only on R, T , ( We want to estimate the error of the Runge Kutta time discretization of the evolution equation (first line of the diagram).To do this, in a first step, we discretize in space by a Galerkin truncation.We estimate the projection error and prove regularity of the solution u m (t) of the projected system (first column in the diagram).In the second step of the proof we investigate the error of the time discretization of the space-discretized system (third row in the diagram) and couple the spatial discretization parameter m with the time step size h in suitable way.In the third step of the proof (third column of the diagram) we prove regularity of the space-time discretization and estimate the projection error of the Runge Kutta time discretization.This concludes the proof.
Step 1 (Regularity of solution of the projected system) In a first step we aim to prove regularity of the continuous solution of the projected system u m (t) = φ t m (P m U 0 ) = Φ t m (U 0 ) which will be needed later.For the proof we denote R from (5.20) as R Φ to indicate that it is a bound on Φ t (U 0 ).We will prove that there is some r φ > 0 such that uniformly in U 0 satisfying (5.20) and m ≥ m * , t ∈ [0, T ], where m * ≥ 0 is sufficiently large.Fix δ > 0. Then we have for U 0 satisfying (5.20) and m ≥ m * .Here M = M 0 [R Φ + δ] and we used (2.4) in the second estimate and Lemma 4.2 and (5.20) in the final estimate.This proves (5.22).
Step 2 (Trajectory error of the time discretized projected system) Next we aim to estimate the trajectory error of the time discretization of the projected system.First note that by Theorem 4.3 (with r replaced by 2r φ and consequently R by 4r φ Λ) there is . Moreover, using (3.3), (3.5a) and (3.5b) we obtain the following bound for h ∈ [0, h * ] to be used later: where M = M 0 [4r φ Λ].Now we define the global error of the projected system, for jh ≤ T , We estimate for any U 0 satisfying (5.20) and for all (n + 1 ≤ ρh p+1 m p+1− + (1 + σ Ψ h)E n m (U 0 , h), for some ρ > 0. Due to (5.24), the second lines of (5.26a) and (5.26b) are valid as long as Clearly E 0 m (U, h) = 0, so (5.28) Using (5.28) we can ensure that for nh by possibly reducing h * > 0, and hence that (5.27) holds.
Step 3 (Projection error of numerical trajectory) We now estimate the global projection error of the numerical method.We will prove that for m(h uniformly for initial data U 0 satisfying (5.20).We first establish the required regularity of the numerical trajectory of the projected system: To bound the Y -norm of the Galerkin truncated numerical trajec- for some r ψ > 0.Here r φ is as in (5.22) and we used (2.4) in the second line and (5.28) in the third line.
To prove (5.30) let e j (U 0 ) = (Ψ h ) j (U 0 ) − (Ψ h m ) j (U 0 ) be the truncation error at time jh ≤ T .Then for (n + 1)h ≤ T , where m = m(h), with order constant uniformly in all U 0 satisfying (5.20), as long as Here we used that for U ∈ P m Y, , by (4.5c) (with r replaced by r ψ and R by where m = m(h) and M = M 0 [2r ψ Λ].In the last inequality of (5.36) we used that where From (5.34) we deduce for nh ≤ T , h ∈ [0, h * ] and all U 0 satisfying (5.20) that with m = m(h).Here (5.36) does not apply to e 1 (U 0 ) Y because in general U 0 / ∈ P m Y.But from (4.5d) we see that e 1 (U 0 ) Y = O(m − ).By choosing a possibly bigger m * (and, by virtue of m = h −p/(p+1) , a smaller h * ) we can achieve that e n (U 0 ) Y ≤ r ψ so that the required condition (5.35) is satisfied.This proves (5.30).
Hence, (4.3b), (5.28) and (5.30) prove that as in [13].We rewrite it in the form (1.1) with U = (u 1 , u 2 ) where u = u 1 + iu 2 with In [13] a second order Strang type time discretization is used to discretize (5.40) and a better rate of convergence is observed, namely an order of convergence O(h) in the H 2 -norm for U 0 ∈ H 4 .This is due to the fact that the linear part of the evolution equation (1.1), i.e., U = AU , is integrated exactly by this method.We plan to extend the methods of this paper to splitting and exponential integrators in future work.

Appendix: Trajectory error on general domains
In this appendix we show how to extend the results of this paper to more general domains.We make the following assumption for the nonlinearity B(U ) of the semilinear evolution equation The right hand side of the evolution equation (1.1) is bounded in the Y norm for U ∈ D ∩ D(A) and it is well-known that there exists a differentiable solution Φ t (U ) ∈ Y in this case, see [17] and Theorem 6.1 below.Extending this setting we will in this section consider initial data U 0 ∈ Y with ∈ J − defined as follows: similarly as in (2.7).For our main result, Theorem 6.10 below, we need an additional condition on the nonlinearity B of (1.1).(B2) B : Here we define [x] + = max(x, 0) for x ∈ R. Assumption (B2) is often satisfied for superposition operators, see Lemma 2.9 b), c) and in particular Example 2.4 where the potential V of the semilinear wave equation is only defined on an open subset D of R.
For a subset U of some Hilbert space Y and δ > 0 we denote by a δ-neighbourhood of U. Moreover for any subset U of Y , ∈ I, we define (U ) δ = U δ as a δ-neighbourhood of U in Y .In the following let U ⊆ D , ∈ I, be a nested collection of open sets and δ > 0 be such that We will also frequently use the abbreviation for ∈ J.
To extend Theorem 2.2 (and also Theorem 3.3, see below) to general domains we cover the domain U with open balls of radius δ and apply the corresponding theorems on each ball.To ensure uniformity of the maximal time interval of existence T * we consider initial data in ( U ) δ/2 0 .Theorem 6.1 (Regularity of the semiflow on general domains).Assume (A) and (B1) and choose > 0. Then there is T * > 0 such that with uniform bounds in U ∈ U .The bounds on T * and Φ, depend only on δ from (6.2), R from (6.3), ω from (2.1), and those afforded by assumption (B1).
Proof.The proof is a modification of the proof of Theorem 2.2.Here we let U 0 ∈ U and take R = δ.As before we compute W as fixed point of the map Π from (2.9), but this time we consider Π as map from (e τ T A − 1)U 0 Y + T e ωT M (6.5) and, for 0 < ≤ min(1, ), we estimate the additional term as follows Here we have used Lemma 6.2 below and that )) with N derivatives in the first component and uniform bounds in U 0 ∈ U .This proves (6.4a).
Note that the term (e τ T A − id)U 0 Y in (2.10) can not be made small uniformly in U ∈ U 0 since the operator e tA is not uniformly continuous in t.But we can make that term order O(T ) uniformly in U 0 ∈ U due to Lemma 6.2 below.
The proof of (6.4b) is similar to the analogous result (2.8) on balls, with obvious modifications.
Proof.To prove (3.6a) let U 0 ∈ U .As in the proof of Theorem 3.3 we compute W as fixed point of the map Π from (3.7), but this time we consider Π as a map from ) and we estimate for h ∈ [0, h * ] and h * small enough and independent of U 0 ∈ U .Here we have used Lemma 6.4 below and that U 0 Y ≤ U 0 Y ≤ R for U 0 ∈ U .The other terms of (6.8) are estimated as in (3.8) with R replaced by δ.So Π maps B δ Y s (1U 0 ) to itself and is a contraction for h * small enough.This proves statements (6.7a) and also (6.7b) in the case k = 0.
Note that the term in (6.9) can not made small independent of U ∈ U 0 since the operator (id −haA) −1 is not uniformly continuous in h.But we can make that term order O(h ) uniformly in U 0 ∈ U due to Lemma 6.4 below.
The rest of the proof is similar to the proof of Theorem 3.3.
The following lemma is an adaptation of Lemma 4.1 and Lemma 4.2 to the setting considered in this section: Lemma 6.5 (Regularity of projected semiflow and projection error on general domains).Assume (A) and (B1), let δ > 0 be as in (6.2) and let > 0. Then there is m * ≥ 0 such that for m ≥ m * there exists a projected semiflow Φ m with the properties specified in Theorem 6.1, with uniform bounds in m ≥ m * .Moreover choose T > 0. Then for sufficiently large m * ≥ 0 the following holds: for all U 0 with Φ t (U 0 ) ∈ U , t ∈ [0, T ] (6.15) and for all m ≥ m * we have Φ t m (U 0 ) ∈ D for t ∈ [0, T ], and (4.3b) is true with an order constant that depends only on δ, R from (6.3), T , (2.1) and the bounds afforded by (B1).
Proof.The only modification required to apply Theorem 6.1 is that we need to choose m * (δ) ≥ 0 large enough to be able to apply the contraction mapping theorem on P m Π(W, P m U, h), with Π as in (2.9), see [16].The proof of (4.3b) is similar to the proof of Lemma 4.2, with obvious modifications.Lemma 6.6 (Regularity of projected numerical method and projection error on general domains).Assume (A), (B1), (RK1) and (RK2), let δ > 0 be as in (6.2) and let > 0. Then there is m * ≥ 0 such that W i m , i = 1, . . ., s and Ψ m satisfy (6.7a) and, if ∈ J − , also (6.7b) with uniform bounds in m ≥ m * .Moreover, if ∈ J − , then (4.5c) and (4.5d) hold true for m ≥ m * , with B r replaced by U .The bounds on h * , m * , Ψ m and W m and the order constants depend only on δ, R from (6.3), (3.5), the bounds afforded by assumption (B1) and on a, b as specified by the numerical method.
Proof.The proof is a modification of the proof of Lemma 4.3.To prove (6.7a) and (6.7b) for the projected numerical method we need to choose m * ≥ 0 large enough to be able to apply the contraction mapping theorem on P m Π(W, P m U, h), with Π as in (3.7), see [16].
To prove (4.5c) in this setting, we need estimate the term in the second line of (4.6) differently than in (4.6) because from (B1) we can not guarantee that W i (U, h) ∈ D , i = 1, . . ., s; in particular this is wrong if > L. Therefore we cannot estimate B(W ) in the Y s norm.We proceed as follows: note that, since ∈ J − there is ∈ (0, 1] such that − ∈ I. Then by (6.7a), with D replaced by D − , there is h * > 0 such that for h ∈ [0, h * ], W i (•, h) ∈ C b ( U , D − ), i = 1, . . ., s. Hence with an order constant uniform in U ∈ U .Here we used (6.16) which will be proved in Lemma 6.7 below.Then solving (4.6) for W (U, h) − W m (U, h) Y s gives (4.5c).
To prove (4.5d) in this setting we estimate the term Q m bh(id −haA) −1 B(W ) Y s in the first line of (4.7) as follows: Inserting this into (4.7)proves (4.5d), with B r replaced by U .

. 6 )
We are now ready to formulate our condition on the nonlinearity B(U ) of (1.1).(B) There exists L ≥ 0, I ⊆ [0, L], 0, L ∈ I, N ∈ N, N > L , such that B ∈ C N − b (B R ; Y ) for all ∈ I and R > 0. We denote the supremum of B : B R → Y as M [R] and the supremum of its derivative as M [R], and set M [R] = M 0 [R] and M [R] = M 0 [R].Moreover we define I − := { ∈ I, − k ∈ I, k = 1, . . ., }.
1) under truncation to a Galerkin subspace of Y.As before for m > 0 we denote by P m the spectral projection operator of A on to the set spec(A) ∩ B m C (0), and set Q m = id −P m .In this setting we define B m (u m ) = P m B(u m ), and consider the projected semilinear evolution equation du m dt = Au m + B m (u m ) (4.1) with flow map φ t m (u 0 m ) = u m (t) for u m (0) = u 0 m ∈ P m Y. Moreover we define Φ t m := φ t m •P m .The Galerkin truncated semiflow has the same regularity properties as the full semiflow (see Theorem 2.2) uniformly in m.

Figure 1 .
Figure 1.Plot of a numerical estimate of q( ) against for the implicit midpoint rule applied to the semilinear wave equation, with the prediction of Theorem 5.3 for comparison.

( 5 . 16 )
Now we show inductively the second estimate of (5.7) and estimate (5.13) for k = , . . ., N .If ∈ N 0 then the start of the induction is k = , and the required estimates are given by Theorem 4.3.If / ∈ N 0 , then the start of the induction is k = > .If k = then, due to (5.16), the first term in (5.15) is of order O(m k− ), and all other terms in the sum of(5.15)  are bounded due to (3.5d) and (5.8) except when j = k in the sum.Hence, using (3.5b), , a, b from the numerical method and the bounds afforded by (B).Proof of Theorem 5.3.The proof consists of several steps, as outlined in the diagram below: Solution of the PDE Error to be estimated RK solution of PDE Projection error ↓ Projection error ↑ Solution of projected PDE −→ RK solution of projected PDE Numerical scheme error .27) Moreover the first supremum of (5.26b) is O(m p+1− ) by Lemma 5.1, with R replaced by r φ .The second supremum of (5.26b) is O(m p+1− ) by Lemma 5.2, with B r replaced by B r φ (and R replaced by 2r φ Λ).

8 ,
and consider it on Y = H 2 (R 3 ; R 2 ).By Lemma 2.9 a) the nonlinearity B(U ) is analytic on Y and the same holds true on Y = D(A ) = H 2( +1) (R 3 , R 2 ) where ≥ 0. In this case assumption (B) holds for I = [0, L] and any L > 0. If (5.40) is discretized by the implicit mid point rule and U 0 ∈ Y 1 = H 4 , then from Theorem 5.3 we obtain an order of convergence O(h 2/3 ) in the H 2 -norm.
(1.1): (B1) There exists L ≥ 0, I ⊆ [0, L], 0, L ∈ I, N ∈ N, N > L and a nested collection of open, Y -bounded sets D ⊂ Y , ∈ I, such that B ∈ C N − b (D ; Y ) for ∈ I. Similarly as before we denote the supremum of B : D → Y as M and the supremum of its derivative as M , and set M = M 0 , M = M 0 and D = D 0 .