Two Modified Runge-kutta Schemes For Two Benjamin-Bona-Mahony-Burgers Equations

Numerical solution of Partial Differential Equations plays an important role in the study of computational mathematics and widely apply to solving many scientific and engineering problems.There are three main methods in the numerical solution of Partial Differential Equations: finite difference method,finite element method and spectral method.The finite difference scheme widely applies to many problems of partial differential equations,for example,heat conduction and so on.The main idea of finite difference scheme is as following: we discrete differential equation with the continuous variant and boundary conditions,and use the approximate solutions for the difference equations as the numerical solutions of the partial differential equations.In the paper,we first presents a modified Runge-Kutta scheme(MRK)for the Benjamin-Bona-Mahony-Burgers(BBM-Burgers)equation.The MRK method combine the finite difference scheme and the four-order Runge-Kutta method.Different finite difference formula is applied to discrete equation in space,and make equation to change a difference equations of differentiating for t,and finally use fourth-order Runge-Kutta method to get the approximate solution.Afterwards,we popularize the MRK scheme to compute generalized BBM-Burgers equation.Finally,we demonstrate that the MRK scheme is capable of higher accuracy,shorter CPU time and good stability by computing some numerical examples.When the parameters of equation change,the MRK scheme still can compute and can get a solution of small error.Evidently,the MRK scheme is an efficient method.