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Fully discrete approximation of the semilinear stochastic wave equation on the sphere

David Cohen, Stefano Di Giovacchino, Annika Lang

Abstract

The semilinear stochastic wave equation on the sphere driven by multiplicative Gaussian noise is discretized by a stochastic trigonometric integrator in time and a spectral Galerkin approximation in space based on the spherical harmonic functions. Strong and almost sure convergence of the explicit fully discrete numerical scheme are shown. Furthermore, these rates are confirmed by numerical experiments.

Fully discrete approximation of the semilinear stochastic wave equation on the sphere

Abstract

The semilinear stochastic wave equation on the sphere driven by multiplicative Gaussian noise is discretized by a stochastic trigonometric integrator in time and a spectral Galerkin approximation in space based on the spherical harmonic functions. Strong and almost sure convergence of the explicit fully discrete numerical scheme are shown. Furthermore, these rates are confirmed by numerical experiments.
Paper Structure (11 sections, 11 theorems, 89 equations, 8 figures)

This paper contains 11 sections, 11 theorems, 89 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Setting and Wiener process on the sphere
  4. The semilinear stochastic wave equation on the sphere
  5. Well-posedness and regularity for the exact solution
  6. Spatial semi-discretization
  7. Temporal discretization
  8. Numerical experiments
  9. The additive case
  10. The multiplicative case
  11. Acknowledgements

Key Result

Lemma 1

For any $s,t\in\mathbb R$, $u\in H^s(\mathbb{S}^2)$ and $v \in H^{s-1}(\mathbb{S}^2)$, the elements of the group matrix $E(t)$ are bounded by These bounds yield the group estimate

Figures (8)

  • Figure 1: Sample paths (at time $0$ (left) and at time $1$ (right)) of the solution to the stochastic wave equation on the sphere with additive noise \ref{['eigen_q']}, nonlinearity \ref{['op_test1']} and initial data \ref{['in_data1']} with $\beta=1$. Here, we take $\alpha=2+1e-6$.
  • Figure 2: Convergence in space: Strong errors (on the left) and pathwise errors (on the right) for different values of $\beta$ and $\delta$ and for the SPDE \ref{['stoc_wave']} with additive noise represented by \ref{['eigen_q']}, nonlinearity \ref{['op_test1']} and initial values \ref{['in_data1']}.
  • Figure 3: Convergence in time: $L^2(\Omega; L^2(\mathbb{S}^2))$, resp. $L^2(\Omega; H^{-1}(\mathbb{S}^2))$ errors, for position, resp. velocity, for different values of $\beta$ and $\delta$ and for the SPDE \ref{['stoc_wave']} with additive noise represented by \ref{['eigen_q']}, nonlinearity \ref{['op_test1']} and initial values \ref{['in_data1']}.
  • Figure 4: Convergence in time: Pathwise $L^2(\mathbb{S}^2)$ and $H^{-1}(\mathbb{S}^2)$ errors for position (on the left) and velocity (on the right) for $\beta=1$, $\delta=1/4$ and for the SPDE \ref{['stoc_wave']} with additive noise represented by \ref{['eigen_q']}, nonlinearity \ref{['op_test1']} and initial values \ref{['in_data1']}.
  • Figure 5: Convergence in time: $L^2(\Omega; L^2(\mathbb{S}^2))$, resp. $L^2(\Omega; H^{-1}(\mathbb{S}^2))$ errors for position, resp. velocity for $\beta=\delta=1$ and for the SPDE \ref{['stoc_wave']} with additive noise represented by \ref{['eigen_q']}, nonlinearities from \ref{['testOp2']} and initial valuers \ref{['in_data1']}.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 8 more