A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise

We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDE) driven by additive space-time noise. We introduce a new modified scheme using a linear functional of the noise with a semi--implicit Euler--Maruyama method in time and in space we analyse a finite element method (although extension to finite differences or finite volumes would be possible). We prove convergence in the root mean square $L^{2}$ norm for a diffusion reaction equation and diffusion advection reaction equation. We present numerical results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation. We see from both the analysis and numerics that the proposed scheme has better convergence properties than the standard semi--implicit Euler--Maruyama method

Efficient numerical schemes for porous media flow

Partial di erential equations (PDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Due to the uncertainty in the input data, stochastic partial di erential equations (SPDEs) have become popular as a modelling tool in the last century. As the exact solutions are unknown, developing e cient numerical methods for simulating PDEs and SPDEs is a very important while challenging research topic. In this thesis we develop e cient numerical schemes for deterministic and stochastic porous media ows. More schemes are based on the computing of the matrix exponential functions of the non diagonal matrices, we use new e cient techniques: the real fast L eja points and the Krylov subspace techniques. For the deterministic ow and transport problem, we consider two deterministic exponential integrator schemes: the exponential time di erential stepping of order one (ETD1) and the exponential Euler midpoint (EEM) with nite volume method for discretization in space. We give the time and space convergence proof for the ETD1 scheme and illustrate with simulations in two and three dimensions that the exponential integrators are e - cient and accurate for advection dominated deterministic transport ow in heterogeneous anisotropic porous media compared to standard semi implicit and implicit schemes. For the stochastic ow and transport problem, we consider the general parabolic SPDEs in a Hilbert space, using the nite element method for discretization in space (although nite di erence or nite volume can be used as well). We use a linear functional of the noise and the standard Brownian increments to develop and give convergence proofs of three new e cient and accurate schemes for additive noise, one called the modi ed semi{ implicit Euler-Maruyama scheme and two stochastic exponential integrator schemes, and two stochastic exponential integrator schemes for multiplicative and additive noise. The schemes are applied to two dimensional ow and transport