A stochastic collocation method for the second order wave equation with a discontinuous random speed

Abstract

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.

Appendix

Lemma 10

Consider the 1D Cauchy problem for the scalar wave equation in the conservative form

$$\begin{aligned} u_{tt} - \partial _x \bigl ( c(x) \, \partial _x u \bigr ) = f(t,x), \qquad (t,x) \in (0,\infty ) \times \mathbb{R }, \end{aligned}$$

(52)

and in the non-conservative form

$$\begin{aligned} u_{tt} - c(x) \, \partial _{xx} u = f(t,x), \qquad (t,x) \in (0,\infty ) \times \mathbb{R }, \end{aligned}$$

(53)

subjected to the initial conditions

$$\begin{aligned} u(0,x) = g(x), \qquad u_t(0,x) = h(x). \end{aligned}$$

Suppose that

• $c(x)$ is positive bounded away from zero and smooth everywhere except at $x=0$ where it has a discontinuity,

• $g(x)$ and $h(x)$ are smooth, compactly supported functions and $0 \notin \text{ supp} \, g \cup \text{ supp} \, h$,

• $\partial _t^k f \in L^2(\mathbb{R })$ for each fixed $t$, and $\partial _t^k f = 0$ at $t=0$ for all $k \ge 0$.

Then, for each fixed $t$, for solutions $u$ to any of the two wave equations (52) and (53),

$$\begin{aligned} \partial _t^k u_t \in L^2(\mathbb{R }), \qquad \partial _t^k u_x \in L^2(\mathbb{R }), \qquad \forall k \ge 0. \end{aligned}$$

Proof

Let $v := \partial _t^k u$. Then, for the conservative form, $v$ solves the Cauchy problem

$$\begin{aligned} v_{tt} - \partial _x \bigl (c(x) \, \partial _x v \bigr ) = \partial _t^k f, \qquad (t,x) \in (0,\infty ) \times \mathbb{R }, \end{aligned}$$

(54)

with the initial conditions

$$\begin{aligned} v(0,x) = \partial _t^k u(0,x) = \left\{ \begin{array}{ll} (\partial _x c \, \partial _x)^{k/2} g,&\quad k \, \text{ even},\\ (\partial _x c \, \partial _x)^{(k-1)/2} h,&\quad k \, \text{ odd}, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} v_t(0,x) = \partial _t^{k+1} u(0,x) = \left\{ \begin{array}{ll} (\partial _x c \, \partial _x)^{k/2} h,&\quad k \, \text{ even},\\ (\partial _x c \, \partial _x)^{(k+1)/2} g,&\quad k \, \text{ odd}. \end{array} \right. \end{aligned}$$

For the non-conservative form, $v$ solves the Cauchy problem

$$\begin{aligned} v_{tt} - c(x) \, \partial _{xx} v = \partial _t^k f, \qquad (t,x) \in (0,\infty ) \times \mathbb{R }, \end{aligned}$$

(55)

with the initial conditions

$$\begin{aligned} v(0,x) = \partial _t^k u(0,x) = \left\{ \begin{array}{ll} (c \, \partial _{xx})^{k/2} g,&\quad k \, \text{ even},\\ (c \, \partial _{xx})^{(k-1)/2} h,&\quad k \, \text{ odd}, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} v_t(0,x) = \partial _t^{k+1} u(0,x) = \left\{ \begin{array}{ll} (c \, \partial _{xx})^{k/2} h,&\quad k \, \text{ even},\\ (c \, \partial _{xx})^{(k+1)/2} g,&\quad k \, \text{ odd}. \end{array} \right. \end{aligned}$$

Since the functions $g$ and $h$ are smooth and their support does not include the discontinuity point of $c(x)$, the initial data for $v$ in both problems are smooth for all $k$. It is well known that for the wave equations (54) and (55) with smooth initial data and $L^2$ forcing term [11],

$$\begin{aligned} v_t, \, v_x \in L^2(\mathbb{R }). \end{aligned}$$

This completes the proof. $\square$

Theorem 7

Consider the 1D Cauchy problem for the scalar wave equation in the conservative form

$$\begin{aligned} u_{tt} - \partial _x\bigl ( c(x,y) \, \partial _x u \bigr ) = f(x), \qquad (t,x,y) \in (0,\infty ) \times \mathbb{R } \times \mathbb{R }, \end{aligned}$$

(56)

subjected to the initial conditions

$$\begin{aligned} u(0,x,y) = g(x), \qquad u_t(0,x,y) = h(x). \end{aligned}$$

(57)

Let $z_{k,l} := \partial _y^k \partial _t^l c u_x$, and assume that the assumptions of Lemma 10 hold. If $\partial _y^k c \in L^{\infty }(\mathbb{R }), \forall k \ge 0$, then for each fixed $t$ and $y$,

$$\begin{aligned} \partial _t z_{k,l} \in L^2(\mathbb{R }), \qquad \partial _{xx} z_{k,l} \in L^2(\mathbb{R }), \qquad \forall k, l \ge 0. \end{aligned}$$

(58)

Proof

We show the result by induction on $k$.

Case $k=0$. We have $\partial _t z_{0,l} = c \, \partial _t^{l+1} u_x$ which belongs to $L^2(\mathbb{R })$ by Lemma A2 for all $l \ge 0$. Moreover, differentiating (56) $l$ times with respect to $t$ and once with respect to $x$ and multiplying by $c$, we obtain

$$\begin{aligned} \partial _{tt} z_{0,l} = c \, \partial _{xx} z_{0,l}. \end{aligned}$$

(59)

Therefore, $\partial _{xx} z_{0,l} \in L^2(\mathbb{R })$ for all $l \ge 0$, because $\partial _t z_{0,l+1} \in L^2(\mathbb{R })$.

General case. We assume that (58) holds with $k < K$. Differentiating (59) $K$ times with respect to $y$ gives us

$$\begin{aligned} \partial _{tt} z_{K,l} - c \, \partial _{xx} z_{K,l} = \sum _{k=0}^{K-1} \genfrac(){0.0pt}{}{K}{k} \, \partial _y^{K-k} c \, \, \partial _{xx}z_{k,l}. \end{aligned}$$

Since the right hand side belongs to $L^2(\mathbb{R })$ by the induction hypothesis, Lemma 10 tells us that $\partial _t z_{K,l} \in L^2(\mathbb{R })$ for all $l \ge 0$. Moreover,

$$\begin{aligned} \partial _{xx} z_{K,l} = \frac{1}{c} \, \Bigl ( \partial _t z_{K,l+1} - \sum _{k=0}^{K-1} \genfrac(){0.0pt}{}{K}{k} \, \partial _y^{K-k} c \, \, \partial _{xx} z_{k,l} \Bigr ), \end{aligned}$$

where the right hand side is in $L^2(\mathbb{R })$. This completes the proof. $\square$