Novel Numerical Methods And Qualitative Analysis Of Rosenau-burgers Equation

As a nonlinear mechanical model with convection and diffusion effect,RosenauBurgers equation has important research value in the fields of quantum field theory,communication science,plasma physics and nonlinear optics.However,it is difficult to obtain analytical solutions.Researchers need to use numerical approximation to obtain the numerical solution of the Rosenau-Burgers equation to analyze the dynamic behavior of this type of problem.Two finite difference schemes are proposed for the homogeneous Rosenau-Burgers equation,namely,the conservation Crank-Nicolson(CN)format and the leapfrog format.The nonlinear terms of these two formats are discretized by a given conservation operator.The solvability are obtained by using Browder fixed point theorem,and convergence of these two formats are obtained by using discrete Gronwall inequality.The conservation of mass and the stability of energy by the partial summation formula and the definition of mass and energy.Numerical examples are given to verify the effectiveness and convergence order of the schemes.Three finite difference schemes are proposed for the inhomogeneous RosenauBurgers equation,namely,the exponential scheme,CN scheme,the modified CN scheme,are linearized by delaying the nonlinear term by a time step,and the original problem is transformed into an ordinary differential equation system for solving.The existence,uniqueness,convergence and stability of the modified CN scheme are obtained by using the Cauchy-Schwarz inequality and the discrete Gronwall inequality.Numerical examples verify the effectiveness and accuracy of the methods.