The Research On Efficient Numerical Methods For Several Kinds Of Differential Equations

Differential equation is one of the important branches of mathematics,which can solve many problems related to derivatives and has a wide range of appli-cations in many fields.Only a few differential equations can obtain analytical solutions.This paper mainly studies the numerical solutions of several kinds of delay differential equations and phase field model equations.The main work of this paper is divided into the following four parts.In the first part,we discuss the global superconvergence of the“postpro-cessed”continuous Galerkin solutions for delay differential equations of nonlinear vanishing delay under quasi-graded meshes by several postprocessing techniques.Based on the supercloseness between the continuous Galerkin solution U and the interpolation?_hu of the exact solution u,we improve the global convergence by some postprocessing methods.Numerical examples show the effectiveness of the postprocessing method.In the second part,the discontinuous Galerkin methods are employed to solve the multi-pantograph delay differential equations.We obtain the global conver-gence and local superconvergence on the uniform meshes.If the right-hand term f is singular,the solution of the multi-pantograph delay differential equation will be singular at the initial value of t=0.The discontinuous Galerkin method can also be used to solve the system of the multi-pantograph delay differential equa-tions,and the same conclusion can be obtained.Numerical experiments show the efficiency of the discontinuous Galerkin method for solving the multi-pantograph delay differential equations.In the third part,we propose and analyze first-and second-order linear schemes for solving the mass-conserving Allen-Cahn equation with local and non-local effects(in the double-well potential case),which are based on the combination of the linear stabilizing technique and the exponential time differencing method.It is well known that the classical Allen Cahn equation can not guarantee the conser-vation property(such as conservation of mass)of the physics described by phase variables after integration over the whole region.Here we introduce Lagrange multipliers with local and nonlocal properties to make the modified Allen-Cahn equation satisfy the conservation of mass.Finally,it is proved that the proposed schemes unconditionally preserve the properties of MBP and mass conservation in time discretization,and the error estimates of these schemes are derived.Numer-ical experiments were carried out to verify the theoretical results.In the fourth part,we formulate a thermodynamic consistent mathematical model to describe mult component phase flow based on the Peng-Robinson equa-tion of state,which is widely used in petroleum industry.It also accurately rep-resent the thermodynamic properties of hydrocarbon mixtures in the multi-phase fluid flow.Since the structure of its energy functional is highly nonlinear and more complicated than many conventional phase field models of the diffuse inter-face model,it is desired to design an accurate and efficient energy stable numerical method for Peng-Robinson equation of state.We analyze and design the first and second order ETD schemes for the diffusion interface model of a single-component two-phase fluid system.The ETD schemes provide a systematic coupling of the explicit treatment involving the nonlinear terms and the implicit and possibly exact integration of the stiff linear parts of the equations,obtaining high accura-cy and maintaining good stability.By introducing the Lagrange multiplier,the Peng-Robinson equation of state maintains the mass conserving,combining with the stabilizer constant,the equivalent equation was proven to possesses the MBP property.Finally,the convergence analysis of the discrete first and second order ETD schemes are derived,and the energy stability of the numerical approximation of the Peng-Robinson equation of state are proved.Numerical experiments were carried out to verify the theoretical results.