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

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.

💡 Research Summary

The paper addresses the numerical approximation of a semilinear stochastic wave equation (SWWE) posed on the unit sphere S² with multiplicative Gaussian noise. While the numerical analysis of SPDEs on Euclidean domains is well‑developed, only a few works have treated SPDEs on manifolds, and none have considered a semilinear wave equation with multiplicative noise on the sphere. The authors fill this gap by proposing a fully discrete scheme that combines a spectral Galerkin method in space (based on spherical harmonics) with an explicit stochastic trigonometric integrator (STI) in time, and they provide rigorous convergence results.

Mathematical setting.
The Laplace–Beltrami operator Δ_{S²} has eigenpairs (−ℓ(ℓ+1), Y_{ℓ,m}) with ℓ∈ℕ₀ and m=−ℓ,…,ℓ. The Q‑Wiener process W(t) is expanded in the same basis, with covariance eigenvalues A_ℓ (angular power spectrum). The SPDE reads ∂{tt}u(t)=Δ{S²}u(t)+f(u(t))+g(u(t))·Ẇ(t), t∈(0,T], with globally Lipschitz nonlinearities f,g. By introducing the state vector X(t)=(u(t),∂_t u(t))ᵗ, the equation is rewritten as a stochastic evolution problem dX(t)=A X(t)dt+F(X(t))dt+G(X(t))dW(t), where A generates a C₀‑group of cosine and sine operators.

Assumptions.
The nonlinearities satisfy a global Lipschitz condition and a linear growth bound in L²(S²). Moreover, the noise term must fulfill ‖g(u) Q^{1/2}‖{L²(H^{δ−1})} ≤ L₂(1+‖u‖{L²}), so that for δ≥1 the noise is trace‑class; for δ∈(0,1) it is only Hilbert‑Schmidt. Initial data are assumed to belong to L^{p}(Ω;H^β) for some β≥0.

Well‑posedness.
Theorem 5 establishes existence, uniqueness, and moment bounds for the mild solution X(t) under the above assumptions, using standard fixed‑point arguments in the product space H^s×H^{s−1}.

Spatial discretisation.
A spectral Galerkin projection P_N onto the span of spherical harmonics with ℓ≤N is applied. The semi‑discrete solution X_N(t) satisfies the same evolution equation with A, F, G replaced by their projected counterparts. Lemma 1–2 provide operator bounds for the cosine and sine semigroups on Sobolev spaces, crucial for error analysis. Theorem 9 proves a strong convergence rate ‖X(t)−X_N(t)‖_{L²(Ω;H)} ≤ C N^{−α}, where α = min(β,δ,1). Thus, higher regularity of the initial data or smoother noise yields faster spatial decay.

Temporal discretisation.
The stochastic trigonometric integrator advances the solution over a uniform time step Δt by exactly propagating the linear wave part: C(Δt)=cos(Δt (−Δ_{S²})^{1/2}), S(Δt)=sin(Δt (−Δ_{S²})^{1/2}), and then adding explicit Euler–Maruyama updates for the nonlinear drift and diffusion. The method is explicit, avoids solving linear systems, and preserves the oscillatory structure of the wave equation. Theorem 11 establishes a strong convergence order ‖X_N(t_n)−X_{N,Δt}^n‖_{L²(Ω;H)} ≤ C (Δt)^β, with β∈(0,½] depending on the noise regularity and the Lipschitz constants. The proof relies on discrete Grönwall inequalities and the operator estimates from Lemma 2.

Almost‑sure convergence.
Using the Borel–Cantelli lemma and the derived moment bounds, the authors also obtain pathwise convergence: for any ε>0, ‖X(t_n)−X_{N,Δt}^n‖_{H} = O(N^{−α+ε}+Δt^{β−ε}) a.s.

Numerical experiments.
Section 5 presents two test problems on S². The first uses a cubic nonlinearity f(u)=−u³ and a multiplicative diffusion g(u)=u; the second adopts additive noise g=I. The covariance operator is chosen with eigenvalues A_ℓ≈ℓ^{−2} (trace‑class) and also with slower decay to illustrate non‑trace‑class effects. The experiments compute the mean‑square error over many Monte‑Carlo samples for varying N and Δt. Log‑log plots confirm the predicted spatial rate α≈1 and temporal rate β≈½. Moreover, sample paths exhibit almost‑sure convergence, matching the theoretical pathwise result.

Contributions and impact.
The paper delivers the first rigorous analysis of a fully discrete scheme for a semilinear SPDE with multiplicative noise on a curved manifold. By exploiting the orthonormality of spherical harmonics, the spatial discretisation handles both the deterministic operator and the stochastic forcing uniformly. The stochastic trigonometric integrator preserves the wave structure and remains explicit, which is advantageous for large‑scale simulations. The results open the way for high‑fidelity stochastic modelling on spherical domains, relevant to climate dynamics, geophysical fluid flows, and astrophysical wave phenomena. Future work may extend the framework to higher‑dimensional manifolds, non‑Lipschitz nonlinearities, or Lévy‑type jumps, and to adaptive or locally refined spectral methods.