Revisionist integral deferred correction methods with application to the moving method of lines

Revisionist integral deferred correction (RIDC) methods are time parallel predictor-corrector methods used to solve initial value problems (IVPs). The prediction and the correction formulae are designed in such a way so that the prediction and the correction steps can be computed simultaneously. More than one computing core can be used at a time to correct the approximate solutions at different correction levels. The multi-core implementation can improve the efficiency of the methods in terms of runtime. In our thesis, we ultimately wish to solve parabolic partial differential equations (PDEs) numerically by combining the spatial adaptive moving mesh method with the time parallel revisionist integral deferred correction (RIDC) method. To do so, we expand an existing RIDC library to handle systems of IVPs of the form L(t, y)y′ = f(t, y), y ∈ Rn. Discretization of a physical PDE by the moving method of lines coupled with a semi-discretized moving mesh PDE results in a system of IVPs of the form L(t, y)y′ = f(t, y), where y(t) is a vector consisting of the physical solution u ∈ Rn and the mesh x ∈ Rn, and L(t, y) is a state dependent square matrix. We achieve a RIDC implementation for this family of IVPs by systematically expanding the existing RIDC formulation and software. We have verified our derived formulae and software with relevant examples