A Finite Difference Method With High-Order Artificial Boundary Conditions For Burgers' Equation On Unbounded Domains

This paper mainly studies the finite difference method using high-order artificial boundary conditions for Burgers' equation on unbounded domains.The first chapter of this paper introduces the research value of Burgers' equation,the common methods to solve problems on unbounded regions,and the application of artificial boundary conditions.In the second chapter,we study the numerical solution of the one-dimensional Burgers' equation in the unbounded region by using the integral artificial boundary conditions.The original problem is converted by Hopf-Cole transformation to the heat equation in the unbounded domain,the latter is reduced to an equivalent problem in a bounded computational domain with two artificial integral boundary conditions,a finite difference method is constructed for the last problem by the method of reduction of order,and therefore the numerical solution of Burgers' equation is obtained.The method is proved and verified to be uniquely solvable,unconditionally stable and convergent with the order 2 in space and the order 3/2 in time for solving the heat equation as well as Burgers' equation in the computational domain.In the third chapter,the numerical solution of Burgers' equation in an unbounded region is studied by using high-order artificial boundary conditions.First,the original problem is converted into the heat equation on an unbounded domain by Hopf-Cole transformation.Thus the difficulty of nonlinearity of Burgers' equation is overcome.Second,high-order artificial boundary conditions are given by using Pade approximation and Laplace transformation.And the conditions confine the heat equation onto a bounded computational domain.Third,we prove the solutions of the resulting heat equation and Burgers' equation are both stable.And the stability and the convergence in an energy norm are also proved theoretically in semi-discrete case.Fourth,the finite difference method using high-order artificial boundary conditions is constructed for the resulting heat equation and the Burgers' equation.Finally,a numerical example demonstrates the stability,the effectiveness and the second-order convergence of the proposed method.