Study On A Class Of Hybrid High Order Non-oscillation Compact Difference Schemes For Typical Partial Differential Equations With Discontinuous Solutions

In this dissertation,a combination scheme of high-order compact difference scheme and a splitting method that preserves accuracy for solving the multi-dimensional viscous Burgers equation was first studied.Secondly,we further explored based on the knowledge base of the combined algorithm,focusing on a hybrid non-oscillation compact difference algorithm for solving degenerate parabolic equations.Finally,the hybrid non-oscillation compact difference algorithm is extended to solve the convection-diffusion equation.For the multi-dimensional Burgers' equations,because the analytical solutions usually involve an infinite series with specific initial and boundary conditions especially when the Reynolds number is large,the solutions of this type of equations converge slowly,and the equations tend to become a convection-dominated problem.Generally speaking,it is necessary to overcome the convection dominated problem.Fortunately,the compact difference schemes combined with the linearized schemes preserving splitting accuracy are used to solve various cases under the condition of large Reynolds number.For the degenerate parabolic equations,they have the characteristic of finite perturbation propagation,which includes discontinuous and strong-shock phenomena,and also have a second-order spatial derivative term,so far there are few researches on solving the degenerate parabolic equations.The essential non-oscillatory scheme and the weighted essential non-oscillatory scheme can well simulate the solution at the discontinuity,but the scheme has the characteristics of excessive dissipation and the resolution needs to be further improved.However,the compact difference scheme has the characteristics of high accuracy,close to the resolution of spectral method and low dissipation.In this paper,a class of hybrid high-order non-oscillatory compact difference method is proposed to solve degenerate parabolic equations.The hybrid high-order non-oscillation compact difference algorithm has high-order,high-resolution,non-physical oscillation properties,which has greater advantages compared to the traditional WENO schemes.For problems involving non-negative physical variables,we have added a positive-preserving limiter.Based on the results of numerical experiments,the effectiveness and applicability of the hybrid high-order non-oscillation compact difference algorithm had proved.And the numerical convergence orders are consistent with the theoretical analysis,which further shows that the hybrid algorithm we proposed is feasible.The paper is divided into five chapters,and the main contents are as follows:In Chapter 1,we give a brief overview of the research background and development of the Burgers' equations and degenerate parabolic equations,and briefly introduce the basic knowledge of compact difference method,ENO and WENO schemes.In Chapter 2,we mainly introduce the compact difference scheme and operator splitting method for the multi-dimensional Burgers' equations.Firstly,the Burgers' equations are linearized and then solved by a compact difference scheme.For two and three-dimensional problems,the operator splitting technique is used to divide them into one-dimensional sub-equation systems and then solve them as well as solve one-dimensional cases.It is theoretically verified that the splitting has no loss of accuracy after linearization.At the same time,compared with the results obtained by directly solving the multi-dimensional Burgers' equations with the compact difference scheme,the numerical results show that the numerical solutions obtained by applying the compact difference scheme to the linearized multi-dimensional Burgers' equation are more accurate.And the effectiveness of the algorithm in this chapter is also verified by numerical examples.In Chapter 3,we mainly design a hybrid high-order non-oscillation compact difference scheme by coupling high-order compact interpolation schemes and odd-even-order WENO interpolation schemes as well as hybrid compact difference methods,which are applied to solve the degenerate parabolic equations.At the same time,the two-dimensional degenerate parabolic equations are solved by combining the operator splitting technique in Chapter 2.In Chapter 4,expanding based on Chapter 3,the hybrid high-order non-oscillation algorithm is applied to some convection-diffusion equations with discontinuous solutions.This chapter explains in detail how to discretize the first-order derivative to obtain higher numerical accuracy.Finally,the results of numerical examples show that the method has high accuracy,high resolution,good robustness,and feasibility.In Chapter 5,we mainly summarizes the key work,innovations and areas to be broken through in the thesis,and also points out some points that can be further studied in this thesis,so as to continue the in-depth discussion in the future.