Numerical approximation and simulation of the stochastic wave equation on the sphere

Abstract

Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrödinger equation on the unit sphere.

Euler–Maruyama approximations of the stochastic heat equation on the sphere

“…Similarly to the proof of Proposition 4.1, we observe first by partial integration that |e −ℓ(ℓ+1)h − (1 + ℓ(ℓ + 1)h) −1 | = −(ℓ(ℓ + 1))2 1 + ℓ(ℓ + 1)h −ℓ(ℓ+1)r dr ds .Using again that x η e −x ≤ Cη to bound (ℓ(ℓ + 1)) 1−µ e −ℓ(ℓ+1)r ≤ C1−µ r µ−1 , we compute the integrals to obtain dr ds = (1 + µ) −1 h 1+µ .Putting these all together yields−(ℓ(ℓ + 1)) 2 1 + ℓ(ℓ + 1)h h 0 h s e −ℓ(ℓ+1)r dr ds ≤ C1−µ (ℓ(ℓ + 1)) 1+µ 1 + ℓ(ℓ + 1)h (1 + µ) −1 h 1+µ = C µ (ℓ(ℓ + 1)) 1+µ h 1+µEM APPROXIMATIONS OF THE STOCHASTIC HEAT EQUATION ON THE SPHERE 41for all µ ∈ (−1, 1], which concludes the proof of a). Using again the same approach as in Proposition 4.1 witha n − b n = (a − b) n−1 j=0 a n−1−j b j and bounding k−1 j=0 e −ℓ(ℓ+1)h•j (1 + ℓ(ℓ + 1)h) −(k−1−j) ≤ e Cc 1 + C c k e −ℓ(ℓ+1)h•(k−1)in a similar way yields both inequalities in b).…”

“…Numerical methods for SPDEs have been developed and analyzed for more than two decades by now, with works, for example, summarized in the monographs [3,10], but studies on surfaces are still rare. We are only aware of the results on the sphere given in [1,2,4,8,9].…”

“…The last thing to introduce from (1) before being able to solve it is the driving noise. Similarly to [8] and [2], we introduce an isotropic Q-Wiener process by the series expansion, often referred to as the Karhunen-Loève expansion,…”

Temporal approximation of stochastic evolution equations with irregular nonlinearities

“…Besides global Lipschitz continuity, no further regularity assumptions are imposed on the nonlinearity F and noise G. Now, our goal is to show pathwise uniform convergence of contractive time discretisation schemes for such irregular nonlinearities and rough initial data, focusing on the hyperbolic setting. It has been extensively studied in recent years (see [1,2,4,7,[10][11][12][14][15][16][17]19,20,26,29,32,[36][37][38][39]42,53] and references therein). When passing to the parabolic setting (i.e.…”