The Generalized Difference Method And LDG Method For The Symmetric Regularized Long Wave Equations

Symmetric regularized long wave equations is used to describe the propagation of plasma acoustic wave and space charge wave under the weakly nonlinear effect. It is one significant class of partial differential equations with many advantages, which has attracted more and more attention towards of the research of well-posed of solutions and numerical methods.In the first chapter, it is mainly introduced that the physics background and research status quo domestic and abroad, and also the application and basic ideological principles of generalized difference method and LDG method about the symmetric regularized long wave equations.In the second chapter, the generalized difference scheme of the symmetric regu-larized long wave equations is designed by discrete the equivalent divisional format of original equations. Then, by applying the numerical interpolation and elliptic projec-tion operators, the error estimate of numerical solution is estimated. Meanwhile, it proves that the format has conservation laws on maintaining the original equations. Finally, it validates the feature of convergence and conservative laws by numerical simulations.In the third chapter, LDG method is considered for solving the initial boundary value problem of symmetric regularized long wave equations, which contains nonlinear high-order derivatives. It is proved that the L2 stability for general solutions and give a detailed error estimate for smooth solutions. For LDG scheme, we select a suitable numerical flux. Then, the scheme to meet the entropy inequality verifies the stability of the method. By using the properties of projection operator, inverse estimate inequality, Young's inequality, etc., we get the order of convergence is O(hk).