Fast finite integration method with variational limit for multi-dimensional partial differential equations

Abstract

Based on the newly developed meshless Finite Integration Method (FIM) where the Trapezoidal/Simpson rule is employed to construct the integration matrices, the FIM with variational limit (VFIM) is further proposed to improve the performance and efficiency in solving the multi-dimensional partial differential equations (PDEs). In contrast to the traditional FIM, the arbitrary functions generated in the integration process can be eliminated easily by introducing the integration with variational limit, such that the tedious deductions and numerical techniques used in approximating the arbitrary functions are no longer needed. This simpler numerical scheme makes VFIM more tractable and efficient in solving PDEs in engineering. Several numerical examples are given to demonstrate the superiority of the VFIM. It can be found that the VFIM with the Trapezoidal/ Simpson rule runs much faster than FIM without the loss of accuracy.