A Conservative Difference Method For Space-Fractional Burgers Equation

A conservative difference scheme is designed to numerically solve the spatial fractional Burgers equation.The finite difference method is used for the discretization in the spatial direction,in which the central difference scheme is used for the discretization of the nonlinear term and the diffusion term,and the second-order approximation method is used for the discretization of the Caputo fractional derivative.The CrankNocolson scheme is used for the discretization in the time direction,and the fully discrete scheme is obtained,this discrete scheme has conservation property.In the process of solving discrete problems,we use the Newton iterative method to deal with nonlinear terms,so that the nonlinear equations are transformed into linear equations.Because the obtained linear equations are asymmetric,We use the biconjugate gradient stabilized method to solve the problem.The structure of the paper is as follows.Firstly,we review the development history of fractional calculus and fractional differential equations,and introduce the research current of fractional Burgers equation at home and abroad.Secondly,we introduce three commonly used definitions of fractional derivatives and Crank-Nicolson numerical methods,and show the second-order approximation method of Caputo fractional derivatives and to get discrete system of stable biconjugate gradient method.Then,we use the finite difference scheme for numerical discretization of spatial fractional Burgers equation,and prove that the discrete system can be transformed into a conservative form.In other words,the difference scheme is conservative.For the theoretical analysis of the conservative difference scheme,we prove its stability,monotonicity and convergence,and the convergence accuracy is O(h2+τ2),where h represents the spatial step and τ represents the time step.Finally,by three numerical examples,we show that the conservative difference scheme is effective on solving the spatial fractional Burgers equation.The first numerical example shows that the convergence order of the numerical algorithm is O(h2+τ2),which is consistent with the theoretical analysis.For the solution of spatial fractional Burgers equation with ’singular solution’,the following two numerical examples show that the scheme has the ability to capture the singular positions of singular solution.