Meshfree Exponential Integrators

For the numerical solution of time-dependent partial differential equations, a class of meshfree exponential integrators is proposed. These methods are of particular interest in situations where the solution of the differential equation concentrates on a small part of the computational domain which may vary in time. For the space discretization, radial basis functions with compact support are suggested. The reason for this choice are stability and robustness of the resulting interpolation procedure. The time integration is performed with an exponential Rosenbrock method. The required matrix functions are computed by Newton interpolation based on Leja points. The proposed integrators are fully adaptive in space and time. Numerical examples that illustrate the robustness and the good stability properties of the method are included.