Generalized finite integration method for solving multi-dimensional partial differential equations

In this paper, we generalize the recently developed Finite Integration Method (FIM) for the solutions of high-dimensional partial differential equations. Formulation of this Generalized Finite Integration Method (GFIM) can be derived due to the use of piecewise polynomials in the numerical integrations. The GFIM does not require the strict requirement for uniformly distributed nodal points in the original FIM. This robustness advantage extends the applicability of FIM to solve partial differential equations by using direct Kronecker product. Due to the unconditional stability of numerical integrations, the GFIM is effective and efficient to solve higher dimensional partial differential equations with stiffness. For numerical verification, we construct several 1D to 4D problems with different types of stiffness and make comparisons among existing numerical methods.