Research Of Some Specific Partial Differential Equations On Numerical Solutions

Partial differential equations are often used to describe technical issues such as engineering,physics and mechanics.In the field of natural science,such as chemical reactions,physical heat conduction phenomena,changes in biological morphology.They can be described by a parabolic equation.Long wave equations are as a way of describing nonlinear dispersive waves.They are often used to explain many scientific branches and physical phenomena such as ion waves and submarine waves.The generalized regular long-wave equation plays a very important role in the physical media.First of all,we construct a parallel algorithm for solving the new difference scheme of parabolic equations.This proposes a brand new parallel format In the odd time layer,the left boundary point uses the classical implicit,the inner point uses the CN format,the right boundary point uses the classical display,the even time layer,the left boundary point uses the classical display and the inner point uses the CN format.The classical boundary point of the right boundary is implicit This method is the alternating use of two double-layer difference schemes.One cycle is the process of continuous alternating calculation and the process is repeated itself We prove that the improved algorithm is shown and absolutely stable.The truncation error at the inner point reaches O(?2+h2),Accuracy improved significantly,the effect is better.Secondly,the numerical solution of the generalized long-wave equation is also studied.The nonlinear term of the generalized long-wave equation contains the parameters.When p is two,it was the famous long-wave equation.Aiming at the initial boundary value problem of regular long wave equation.We propose two new conservative difference schemes for the initial boundary value problem of the regularized long wave equations,that is the two-layer linear conservation difference scheme and the three-layer nonlinear conservation difference scheme.Theoretical analysis and numerical simulation experiments verify the feasibility and effectiveness of the new method.Finally,we propose two new conservative difference schemes for the initial boundary value problems of the generalized regular long-wave equations,that is a two-layer linear-conservative difference scheme and a three-layer nonlinear conservation difference scheme.The new format simulates a conservation property of the initial boundary value problem for a generalized long-wave equation and performs a priori estimation of the diferential solution.We conduct a priori estimates of the differential solutions,and make theoretical analysis of the existence of the solutions,the stability and convergence of the difference schemes.Simulation results show that the new method is superior to the previous one,the effect of energy conservation is good.when the appropriate weight coefficient is selected,the computational accuracy will be improved greatly.