Study On Numerical Algorithm Of Several Nonlinear Partial Differential Equations

Partial differential equations are widely used in various fields of science.These partial differential equations can transform engineering problems into mathematical problems.But some partial differential equations solutions are very complicated.In order to solve this kind of problem,people use numerical methods to solve such partial differential equations.And in this case,according to the initial conditions and boundary conditions,people construct discrete scheme.In recent years,scholars,all over the world,are concerned on greatly how to calculate the numerical solution of partial differential equations.In this paper,both the nonlinear modified regular long wave equation and Rosenau equation are studied.The order of the equations are reduced by introducing the parameters.With the multiple integral finite volume method,we construct one type of discrete schemes for the two equations with the initial and boundary value conditions.And we prove the properties concerning the discrete schemes,such as conservation and stability.At the same time,we prove the properties about the numerical solution,such as existence,uniqueness,priori estimation and convergence.Finally,with numerical experiments,energy analysis and error estimation of the discrete schemes are carried out.On the one hand,we study the nonlinear MRLW equation with second-order partial derivative in space direction.By quoting auxiliary variables,we combine the multiple integral finite volume method and Taylor function method to obtain the spatial semi-discrete numerical scheme.And with the Crank-Nicolson method,the time direction of the equation is discretized in time layern+1/2.So far,we obtain a two-level nonlinear implicit fully discrete numerical scheme.The scheme has fourth-order accuracy in space direction and second-order accuracy in time direction.The conservation and stability of the discrete scheme are derived.The existence,uniqueness,priori estimation and convergence about numerical solution are proved.According to some numerical experiments,we analyze the energy conservation and error estimation of this discrete scheme.At the same time,by comparing with other reference,it is proved that the property concerning energy conservation of this scheme is better than other reference.What’s more,the numerical solution of this scheme has minor errors.On the other hand,we study the nonlinear Rosenau equation.For this equation,the fourth-order space partial derivative is reduced to second-order partial derivative by quoting auxiliary variables.It is helpful to construct discrete scheme for the equation.In space direction,we use the multiple integral finite volume method to obtain the semi-discrete sche-me.In time direction,the equation is discretized on time layern+1/2 by the Crank-Nicolson method.The fully discrete scheme is obtained with O(h~4+τ~2)accuracy.In theory aspect,the conservation stability of the scheme is proved strictly and the properties of numerical solutions are derived including existence,uniqueness,priori estimation and convergence.By numerical experiments,we analyze the conservation and error of the discrete scheme and prove that the discrete scheme is effective.