High-precision Numerical Methods For Solving Nonlinear Burgers’Equation

Burgers’ equation is an important and basic parabolic PDE in fluid mechanics. Various numerical schemes for solving Burgers’ equation can be summarized as finite difference, finite element, spectral element, spline approximations and so on. In this paper, the main content is divided into two parts. We study numerical methods for Burgers’ equation in bounded and unbounded domain respectively.In the first part, we study the numerical solution of one-dimensional Burgers’equation with non-homogeneous Dirichlet boundary conditions in bounded domain. This nonlinear problem is converted into the linear heat equation with non-homogeneous Robin boundary conditions by Hopf-Cole transformation. The heat equation is discretized by Crank-Nicolson finite difference scheme, and the fourth-order difference schemes for the Robin conditions are combined with the Crank-Nicolson scheme at two endpoints. The proposed method is proved to be second-order convergent and unconditionally stable. The numerical example supports the theoretical results.In the second part, we construct a numerical method for one-dimensional Burgers’equation in the unbounded domain by using 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 by using two artificial integral boundary conditions, a finite difference method with discrete artificial boundary conditions is established by using the method of reduction of order for the last problem, and thereupon the numerical solution of Burgers’equation is obtained. This artificial boundary method is proved and verified to be uniquely solvable, unconditionally stable and convergent with the order2in space and the order3/2in time for solving Burgers’ equation on the computation domain.