Exponential integrators for stochastic partial differential equations
Sammanfattning: Stochastic partial differential equations (SPDEs) have during the past decades become an important tool for modeling systems which are influenced by randomness. Because of the complex nature of SPDEs, knowledge of efficient numerical methods with good convergence and geometric properties is of considerable importance. Due to this, numerical analysis of SPDEs has become an important and active research field.The thesis consists of four papers, all dealing with time integration of different SPDEs using exponential integrators. We analyse exponential integrators for the stochastic wave equation, the stochastic heat equation, and the stochastic Schrödinger equation. Our primary focus is to study strong order of convergence of temporal approximations. However, occasionally, we also analyse space approximations such as finite element and finite difference approximations. In addition to this, for some SPDEs, we consider conservation properties of numerical discretizations.As seen in this thesis, exponential integrators for SPDEs have many benefits over more traditional integrators such as Euler-Maruyama schemes or the Crank-Nicolson-Maruyama scheme. They are explicit and therefore very easy to implement and use in practice. Also, they are excellent at handling stiff problems, which naturally arise from spatial discretizations of SPDEs. While many explicit integrators suffer step size restrictions due to stability issues, exponential integrators do not in general.In Paper 1 we consider a full discretization of the stochastic wave equation driven by multiplicative noise. We use a finite element method for the spatial discretization, and for the temporal discretization we use a stochastic trigonometric method. In the first part of the paper, we prove mean-square convergence of the full approximation. In the second part, we study the behavior of the total energy, or Hamiltonian, of the wave equation. It is well known that for deterministic (Hamiltonian) wave equations, the total energy remains constant in time. We prove that for stochastic wave equations with additive noise, the expected energy of the exact solution grows linearly with time. We also prove that the numerical approximation produces a small error in this linear drift.In the second paper, we study an exponential integrator applied to the time discretization of the stochastic Schrödinger equation with a multiplicative potential. We prove strong convergence order 1 and 1/2 for additive and multiplicative noise, respectively. The deterministic linear Schrödinger equation has several conserved quantities, including the energy, the mass, and the momentum. We first show that for Schrödinger equations driven by additive noise, the expected values of these quantities grow linearly with time. The exponential integrator is shown to preserve these linear drifts for all time in the case of a stochastic Schrödinger equation without potential. For the equation with a multiplicative potential, we obtain a small error in these linear drifts.The third paper is devoted to studying a full approximation of the one-dimensional stochastic heat equation. For the spatial discretization we use a finite difference method and an exponential integrator is used for the temporal approximation. We prove mean-square convergence and almost sure convergence of the approximation when the coefficients of the problem are assumed to be Lipschitz continuous. For non-Lipschitz coefficients, we prove convergence in probability.In Paper 4 we revisit the stochastic Schrödinger equation. We consider this SPDE with a power-law nonlinearity. This nonlinearity is not globally Lipschitz continuous and the exact solution is not assumed to remain bounded for all times. These difficulties are handled by considering a truncated version of the equation and by working with stopping times and random time intervals. We prove almost sure convergence and convergence in probability for the exponential integrator as well as convergence orders of ½ − ?, for all ? > 0, and 1/2, respectively.
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