A Radial Basis Function - Finite Difference and Parareal Framework for Solving Time Dependent Partial Differential Equations

The Radial Basis Function–Finite Difference (RBF–FD) method is a mesh-less method for discretizing differential operators in ordinary differential equations (ODEs) and partial differential equations (PDEs). To solve a time-dependent PDE, it is common to use the method of lines approach (MOL), where one discretizes the spatial differential operator using RBFs, converting the PDE into a system of ODEs. The system of ODEs is solved using an appropriate numerical ODE solver. The RBF–FD approach has the advantage of leading to a differentiation matrix that is sparse. However, spurious eigenvalues could lead to unstable algorithms and resolving the issue could increase the computational cost. Typically, solving an ODE is usually thought of as a set of sequential steps. In this work, we propose to use a parallel ODE solver called the Parareal method that offers the ability to compute expensive steps in parallel. We also introduce a new strategy to utilize coarse and fine differentiation matrices along with coarse and finer ODE solvers in the Parareal algorithm to further speed up the computations. This strategy also notably mitigates the effects that the spurious eigenvalues have on the computations and reduces the computational cost of solving the system of ODEs with a standard approach.