Exponential Integrators for Diffusion Equations

Sammanfattning

This thesis studies exponential time integration methods for diffusion equations after finite element discretization. Such problems can lead to stiff systems of ordinary differential equations, since the discretized diffusion operator contains rapidly decaying modes. The exponential Euler method and a second-order exponential Runge–Kuttamethod are derived, implemented, and tested. Numerical experiments on stiffmodel problems and a one-dimensional finite element diffusion problem confirm the expected convergence rates: first order for exponential Euler andsecond order for the Runge–Kutta method. For linear homogeneous diffusion, exponential Euler reproduces the exact semi-discrete time evolution upto spatial and numerical errors. The computational cost of evaluating the matrix exponential is also considered. Direct eigendecomposition is accurate for moderate problem sizes, while Krylov subspace methods provide an alternative for larger systems. Overall, the results show that exponential integrators are accurate and well suited fordiffusion-dominated problems, provided that matrix exponential actions can be computed efficiently.