Analysis of exponential splitting methods for inhomogeneous parabolic equations

摘要

We analyze the convergence of the exponential Lie and exponential Strangsplitting applied to inhomogeneous second-order parabolic equations withDirichlet boundary conditions. A recent result on the smoothing properties ofthese methods allows us to prove sharp convergence results in the case ofhomogeneous Dirichlet boundary conditions. When no source term is present andnatural regularity assumptions are imposed on the initial value, we showfull-order convergence of both methods. For inhomogeneous equations, we provefull-order convergence for the exponential Lie splitting, whereas orderreduction to 1.25 for the exponential Strang splitting. Furthermore, we givesufficient conditions on the inhomogeneity for full-order convergence of bothmethods. Moreover our theoretical convergence results explain the severe orderreduction to 0.25 of splitting methods applied to problems involvinginhomogeneous Dirichlet boundary conditions. We include numerical experimentsto underline the sharpness of our theoretical convergence results.