Finite line method for solving high-order partial differential equations in science and engineering

Abstract

In this paper, a completely new numerical method, named Finite Line Method (FLM), is proposed for solving general linear and non-linear high-order partial differential equations (PDEs) in science as well as engineering problems in heat conduction and mechanics. In this method, the computational domain is discretized into a number of collocation points as in the free element method (Gao et al., 2019; Liu et al., 2019), and at each collocation point a set of straight or curved lines crossing the point is formed, which is represented by a number of nodes distributed over lines. The shape functions for each line are constructed using the Lagrange interpolation formulation and the first and high order spatial partial derivatives of the shape functions with respect to global coordinates are derived through an ingenious technique. The derived partial derivatives can be directly substituted into the governing differential equations and related boundary conditions to set up the discretized system of equations.FLM is a type of collocation method, not needing any integration to establish the solution scheme. And since the Lagrange interpolation formulation is used to construct the shape functions, arbitrarily high order lines can be easily formulated for solving any types of linear and non-linear high order PDEs and engineering problems. A number of numerical examples for PDEs, heat conduction and solid mechanics problems are given to demonstrate the correctness and stability of the proposed method.