Local Discontinuous Galerkin Method For Sobolev-type Equations

The Sobolev-type equations are widely used in fields of mathematics,physics and engineering,and the related numerical methods have attracted considerable interest of research scholars.However,the corresponding scheme design and theoretical analysis become especially difficult,because such equations contain mixed derivatives of time and space variables.This dissertation intends to establish the local discontinuous Galerkin(LDG)method for Sobolev-type equations,and to verify the effectiveness of the LDG algorithm from two perspectives of theoretical analysis and numerical experiments.The main highlight of this method is the full application of space-time transformation relationship between auxiliary variables,making the corresponding timemarching simple and efficient.This dissertation includes five chapters.In the first chapter,we present some reviews of development of Sobolev-type equations and the LDG method.In the last chapter,we give some conclusions and on-going work.The second to fourth chapters are the main body of the dissertation,which are organized as follows.In the second chapter,we construct the LDG scheme for the two-dimensional linear Sobolev equation,and give the stability and optimal error estimates of semiand fully-discrete schemes.Here we use generalized numerical fluxes and the thirdorder total variation diminishing explicit Runge-Kutta(TVDRK3)method.We make full use of space-time transformation relationship to transform the high-order timedevelopment equation into the first-order time-development equations.This approach has two numerical advantages:(1)the temporal evolution of numerical solutions can be decoupled as a direct time-updating on degrees of freedom;(2)the definitions of numerical fluxes become clear and natural.In this chapter,we give two definitions of initial values and show their differences: when original and auxiliary variables also satisfy the DG spatial discretization relationship at initial time,the LDG scheme is strongly stable with respect to μ-norm;otherwise,the corresponding scheme is merely stable in weak sense.With the help of the two-dimensional generalized Gauss-Radau(GGR)projection and its modified projection,we coordinate the coupling relationship between different numerical fluxes on each cell boundary,and then obtain a sharp error estimate,which is consistent with experimental results.It’s worth mentioning that the error bounding constant is independent of the reciprocal of high-order term coefficients.In other words,the results in this chapter still hold for the convectiondominated Sobolev-type problems.In the third chapter,we apply the LDG method to MBL equation.This equation is a typically nonlinear Sobolev equation,which contains non-convex convection term and is convection-dominated.The corresponding solution is often accompanied by transient layer that exhibits large gradient change.The purpose of this chapter is to verify effectiveness of the LDG method for solving such problem.A large amount of experiments show that when viscosity coefficient ε of the equation is not very small,the LDG method can capture the transient layer well;However,when ε gradually becomes smaller,numerical solution occurs violent numerical oscillation near the transient layer.To improve the numerical accuracy,we introduce TVB limiter,WENO limiter and Moe limiter,and demonstrate that a suitable Moe limiter leads to more accurate approximation of solution and suppresses the numerical oscillation.In addition,when capturing the structure of transient layer,the global mesh refinement will cause unnecessary calculations in smooth regions.For this purpose,an adaptive moving mesh method is also considered at the end of this chapter to implement local refinement near the transient layer.Numerical experiments shows that to obtain the same accuracy,the moving mesh method effectively reduces the spatial degrees of freedom and improves the calculation efficiency.In the fourth chapter,we apply the LDG method to the other two kinds of equations.One is the linearized coupled BBM system,and the other is the KDV equation with fifth-order mixed derivatives of time and space.These two equations have or can be transformed into Sobolev-type equations.In the first section,we combined the TVDRK3 time-marching to construct the semi-and fully-discrete LDG schemes for coupled BBM system.Using anti-symmetric property of DG spatial discretization based on generalized alternating flux,we can prove that the scheme is strongly stable with respect to L2-norm.At the same time,by means of the one-dimensional GGR projection technique,we obtain the optimal error estimate in L2-norm.In the second section,we discuss the theoretical analysis of the corresponding LDG method to solve the fifth-order KDV equation.The key of this scheme is to introduce appropriate auxiliary variables and time derivation,by which we can convert the fifth-order equation into a Sobolev system of equations,and thus give the LDG method for this system.If the initial value satisfies the DG spatial discretization relationship,the scheme has similar stability as the previous section.Based on this,using one-dimensional GGR projection and its modified projection,a sub-optimal L2-norm convergence order can be proved.