Abstract
In this paper, we consider the linear finite volume method (FVM) for the stochastic Helmholtz equation, driven by white noise perturbed forcing terms in one-dimension. We first deduce the linear FVM for the deterministic Helmholtz problem. The dispersion equation is presented, and the error between the numerical wavenumber and the exact wavenumber is then analyzed. Comparisons between the linear FVM and the linear finite element method (FEM) are also made. The theoretical analysis and practical calculation indicate that the error of the linear FVM is half of that of the linear FEM. For the stochastic Helmholtz equation, convergence analysis and error estimates are given for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are provided to examine our theoretical results.
Highlights
In this paper, we consider the stochastic Helmholtz problem in one-dimension driven by an additive white noise forcing term ⎧ ⎨– d2u dx2 k2u(x) = g(x) +W (x), x ∈ (0, 1),⎩u(0) = 0, u (1) – iku(1) = 0, (1.1)
We find that the term associated with k3h2 for the linear finite volume method (FVM) is half of that for the linear finite element method (FEM), when kh is small enough
6 Conclusions In this paper, we proposed the linear FVM for the stochastic Helmholtz equation, driven by an additive white noise forcing term in one-dimension
Summary
We consider the stochastic Helmholtz problem in one-dimension driven by an additive white noise forcing term (see [6, 15, 18]). To numerically solve the stochastic Helmholtz equation, we should consider two issues: one is randomness, and the other is a high wavenumber. We apply the linear FVM to the stochastic Helmholtz equation with discretized white noise forcing terms. 4, we study the linear FVM of the stochastic Helmholtz equation with discretized white noise forcing terms, and obtain the H1 error estimates between the finite volume solutions and the exact solution of (1.1). Numerical results illustrate that, when solving the deterministic Helmholtz problem, the error for the linear FVM is only half of that for the linear FEM.
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