A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

We investigate a discretization of a class of stochastic heat equations on the unit sphere with multiplicative noises. A spectral method is used for the spatial discretization and the truncation of the Wiener process, while an implicit Euler scheme with non-uniform steps is used for the temporal discretization. Some numerical experiments inspired by Earth's surface temperature data analysis GISTEMP provided by NASA are given.

[1]  Volker Schönefeld.  Spherical Harmonics , 2019, An Introduction to Radio Astronomy.

[2]  Klaus Ritter,et al.  Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations , 2007, Found. Comput. Math..

[3]  K. Ritter,et al.  An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise , 2006, math/0604600.

[4]  Yubin Yan,et al.  Galerkin Finite Element Methods for Stochastic Parabolic Partial Differential Equations , 2005, SIAM J. Numer. Anal..

[5]  V. Bogachev.  Gaussian Measures on a , 2022 .

[6]  T. Shardlow.  Numerical methods for stochastic parabolic PDEs , 1999 .

[7]  G. Peccati,et al.  Random Fields on the Sphere: Spectral Representations , 2011 .

[8]  Qiang Du,et al.  Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises , 2002, SIAM J. Numer. Anal..

[9]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[10]  Zhimin Zhang,et al.  Finite element and difference approximation of some linear stochastic partial differential equations , 1998 .

[11]  E. Hausenblas.  Numerical analysis of semilinear stochastic evolution equations in Banach spaces , 2002 .

[12]  E. Hausenblas.  Approximation for Semilinear Stochastic Evolution Equations , 2003 .

[13]  Karri Muinonen,et al.  Scattering of light by large Saharan dust particles in a modified ray optics approximation , 2003 .

[14]  C. Schwab,et al.  Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations , 2013, 1305.1170.

[15]  I. Gyöngy.  Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II , 1999 .

[16]  B. Veihelmanna,et al.  Light scattering by small feldspar particles simulated using the Gaussian random sphere geometry , 2006 .

[17]  D. Nualart,et al.  Implicit Scheme for Stochastic Parabolic Partial Diferential Equations Driven by Space-Time White Noise , 1997 .

[18]  G. Peccati,et al.  Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications , 2011 .

[19]  P. Kloeden,et al.  Time-discretised Galerkin approximations of parabolic stochastic PDE's , 1996, Bulletin of the Australian Mathematical Society.

[20]  Jessica G. Gaines,et al.  Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations , 2001, Math. Comput..

[21]  Yubin Yan,et al.  Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise , 2004 .

[22]  Greg Michael McFarquhar,et al.  Light Scattering by Quasi-Spherical Ice Crystals , 2004 .

[23]  G. Sell,et al.  Dynamics of Evolutionary Equations , 2002 .

[24]  P. Kloeden,et al.  LINEAR-IMPLICIT STRONG SCHEMES FOR ITO-GALKERIN APPROXIMATIONS OF STOCHASTIC PDES , 2001 .

[25]  M. Lifshits.  Gaussian Random Functions , 1995 .

[26]  G. Lord,et al.  A numerical scheme for stochastic PDEs with Gevrey regularity , 2004 .