Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations

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.

Appendix: Stability estimates

In this Appendix we sketch the Proof of Lemma 3.7.

For $\varphi \in {}\overline{\mathcal B}_Y(R)$ we define inner product

$$\begin{aligned} ( x,y )_\varphi = ( \Lambda (\varphi ) x, y )_X. \end{aligned}$$

With $X_\varphi$ we denote the space X endowed with this inner product. From (5a) and (8a) follows that the associated norm is uniformly equivalent to the X-norm, i.e.,

$$\begin{aligned} \lambda _X^{-1} \left\| x\right\| ^2_\varphi \le \left\| x\right\| ^2_X \le \nu \left\| x\right\| ^2_\varphi , \qquad x\in X. \end{aligned}$$

(45)

By using (5c) and (45), for $\varphi ,\psi \in {}\overline{\mathcal B}_Y(R)$, we have

$$\begin{aligned} \left\| x\right\| ^2_\varphi&= ( \Lambda (\varphi )x,x )_X = ( \Lambda (\psi )x,x )_X + ( (\Lambda (\varphi )-\Lambda (\psi ))x,x )_X \\&\le \left\| x\right\| ^2_\psi + \ell \left\| \varphi -\psi \right\| _Y \left\| x\right\| ^2_X \le (1+\ell \nu \left\| \varphi -\psi \right\| _Y) \left\| x\right\| ^2_{\psi }. \end{aligned}$$

It follows that

$$\begin{aligned} \left\| x\right\| _\varphi \le e^{k_1 \tau } \left\| x\right\| _\psi \qquad \text {for} \qquad \left\| \varphi -\psi \right\| _Y\le \gamma \tau , \end{aligned}$$

(46)

where $k_1:=k_1(\gamma )$ is defined in (9a).

For a Banach space V and real numbers $C\ge 1$ 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 $\varphi \in {}\overline{\mathcal B}_Y(R)$ there holds

$$\begin{aligned} A_\varphi \in G(X_\varphi , 1, \omega ), \end{aligned}$$

(47a)

where $\omega$ is defined in (9c), which then implies the following bound for the resolvent

$$\begin{aligned} \left\| (I- \tau A_\varphi )^{-1}\right\| _{X_\varphi \leftarrow X_\varphi } \le (1-\tau \omega )^{-1} \quad \text {for} \quad \tau \omega < 1. \end{aligned}$$

(47b)

From $( \Lambda (\varphi )^{-1}A x,x )_{\varphi } = 0$, Assumption 2.2(b), the fact that A is a closed operator in X (since it is skew-adjoint) and the norm equivalence (45) we can conclude by using the Lumer–Phillips theorem, cf. [6, Theorem II.3.15], that $\Lambda (\varphi )^{-1}A$ generates a contraction semigroup on $X_\varphi$. Further on, $\Lambda (\varphi )^{-1} Q(\varphi )-\omega I$ is a bounded operator on $X_\varphi$ and

$$\begin{aligned} ( (\Lambda (\varphi )^{-1} Q(\varphi )-\omega I)x,x )_\varphi \le \mu _X \left\| x\right\| ^2_X - \omega \left\| x\right\| ^2_\varphi \le (\mu _X \nu -\omega ) \left\| x\right\| ^2_{\varphi }=0, \end{aligned}$$

i.e., $\Lambda (\varphi )^{-1} Q(\varphi )-\omega I$ is dissipative in $( \cdot ,\cdot )_\varphi$. Therefore, by the perturbation result [6, Theorem III.2.7], we have that $A_\varphi -\omega I$ generates a contraction semigroup on $X_\varphi$, i.e. $A_\varphi -\omega I \in G(X_\varphi , 1, 0)$. (47) now follows by the bounded perturbation theorem (cf. [6, Theorem III.1.3]).

We proceed as follows by using (45), (46) and (47) to obtain the X-norm estimate. For $\tau \omega <1$ there holds

$$\begin{aligned}&\left\| (I-\tau A_{\varphi _k})^{-1} \cdots (I-\tau A_{\varphi _j})^{-1} u\right\| _X \\&\quad \le \nu ^{1/2} \left\| (I-\tau A_{\varphi _k})^{-1} \cdots (I-\tau A_{\varphi _j})^{-1} u\right\| _{\varphi _k} \\&\quad \le \nu ^{1/2} (1 - \tau \omega )^{-1}\left\| (I-\tau A_{\varphi _{k-1}})^{-1} \cdots (I-\tau A_{\varphi _j})^{-1} u\right\| _{\varphi _k} \\&\quad \le \nu ^{1/2} (1 - \tau \omega )^{-1} e^{k_1 \tau } \left\| (I-\tau A_{\varphi _{k-1}})^{-1} \cdots (I-\tau A_{\varphi _j})^{-1} u\right\| _{\varphi _{k-1}}\\&\quad \le \cdots \le \nu ^{1/2} (1-\tau \omega )^{-(k-j+1)} e^{k_1 (k-j)\tau } \left\| u\right\| _{\varphi _j}\\&\quad \le k_0 (1-\tau \omega )^{-(k-j+1)} e^{k_1 (k-j)\tau } \left\| u\right\| _X. \end{aligned}$$

To get the Z-norm estimate we use the operator $A_{\varphi }^S = A_\varphi + B(\varphi )$ defined in (7a). For $\varphi \in {}\overline{\mathcal B}_Y(R)\cap {}\overline{\mathcal B}_Z(r)$, by (7b) and (45), we obtain

$$\begin{aligned} \left\| B(\varphi ) x\right\| _\varphi \le \lambda _X^{1/2} \left\| B(\varphi ) x\right\| _X \le \lambda _X^{1/2} \beta \left\| x\right\| _X \le k_0 \beta \left\| x\right\| _\varphi , \end{aligned}$$

i.e., $\left\| B(\varphi )\right\| _{X_\varphi \leftarrow X_\varphi } \le k_0 \beta$. Applying the bounded perturbation theorem again gives that for $\varphi \in {}\overline{\mathcal B}_Y(R)\cap {}\overline{\mathcal B}_Z(r)$ it holds

$$\begin{aligned} A_{\varphi }^S \in G(X_\varphi , 1, \widetilde{\omega }) \end{aligned}$$

where $\widetilde{\omega }$ is defined in (9c). For $\tau \widetilde{\omega }<1$ we can write

$$\begin{aligned}&\left\| (I-\tau A_{\varphi _k})^{-1} \cdots (I-\tau A_{\varphi _j})^{-1} u\right\| _Z\\&\quad = \left\| S^{-1} (I-\tau A^S_{\varphi _k})^{-1} \cdots (I-\tau A^S_{\varphi _j})^{-1} S u\right\| _Z \\&\quad \le \left\| S^{-1}\right\| _{Z \leftarrow X} \left\| (I-\tau A^S_{\varphi _k})^{-1} \cdots (I-\tau A^S_{\varphi _j})^{-1} S u\right\| _X \end{aligned}$$

and proceed as above. This yields the bound in the Z-norm. Since Y is an exact interpolation space between Z and X, the second inequality follows immediately.