Error Estimates And Bound Preserving Schemes Of Discontinuous Galerkin Methods For Nonlinear Equations

In this thesis,we mainly study the discontinuous Galerkin method for nonlinear partial differential equations in bounded domain.We first prove the energy stabil-ity and the optimal error estimates of the local discontinuous Galerkin methods for the Allen-Cahn equation and Cahn-Hilliard equation.Secondly,based on the Karush-Kuhn-Tucker(KKT)limiter,we respectively construct higher order bounds preserv-ing time-implicit discontinuous Galerkin and local discontinuous Galerkin discretiza-tions using Lagrange multipliers for reactive Euler equations and nonlinear degenerate parabolic equations.In the first part,we study the stability and the optimal error estimates of the sec-ond and third order semi-implicit spectral deferred correction(SDC)time discrete local discontinuous Galerkin schemes for the Allen-Cahn equation.Since the SDC method is based on the first order convex splitting scheme,the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step,which increases the diffi-culty of the theoretical analysis.For the local discontinuous Galerkin discretizations coupled with the second and third order SDC methods,we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimen-sional spaces.At the same time,the iteration and integral involved in the semi-implicit SDC scheme also increase the difficulty of the theoretical analysis.Comparing to the Runge-Kutta type semi-implicit schemes which exclude the left-most endpoint,the SDC scheme in this paper includes the left-most endpoint as a quadrature node.This makes the test functions of the SDC scheme are more complicated and the energy equations are more difficult to be constructed.We provide two different ideas to overcome the difficulty of the nonlinear terms.By choosing the test functions carefully,the energy stability and the optimal error estimates are obtained in the sense that the time step ?only requires a positive upper bound and is independent of the mesh size h.Numerical examples are presented to illustrate our theoretical results.In the second part,we mainly study the error analysis of an unconditionally en-ergy stable local discontinuous Galerkin scheme for the Cahn-Hilliard equation with concentration-dependent mobility.The time discretization is based on the invariant energy quadratization(IEQ)method.The fully discrete scheme leads to a linear alge-braic system to solve at each time step.The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the local discontinuous Galerkin discretization.Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn-Hilliard equation.For the error estimates of the initial condition,the nonlinear term is equally replaced by an equivalent smooth and globally Lipschitz continuous function.This technique is only used for the problem at initial time.For the analysis of the non-constant mobility term,we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L?-norm by the mathematical induction method.The optimal error results are obtained for the fully discrete scheme and the numerical results are given to verify this conclusion.In the third part,we construct higher order bounds preserving time-implicit discon-tinuous Galerkin discretizations for the reactive Euler equations.In reaction problems,the time step can be significantly limited because of the big difference between the fluid dynamics time scale and reaction time scale.And the density and pressure are nonneg-ative,the mass fractions should be between zero and one.We use the fractional step method to deal with the convection step and reaction step separately.There are three main contributions to the reactive Euler equations.Firstly,the higher order diagonally implicit Runge-Kutta(DIRK)methods are adopted for time discretization.Compared to the explicit time-discrete methods,the implicit methods greatly enlarge the time step for the equations with stiff source terms.Secondly,depending on the KKT system,the lower bound zero and upper bound one can be preserved by combining the time-implicit numerical discretizations with bounds preserving constraint using Lagrange multiplier.Finally,due to the stiff source terms,we extend the usage of Harten's subcell resolution(SR)technique to time-implicit discontinuous Galerkin methods in the reaction step.Numerical results are shown to demonstrate that the bounds preserving DIRK discon-tinuous Galerkin discretizations are higher order accurate for the smooth solutions and rather efficient on relatively coarse meshes for the stiff problems with discontinuities.In the forth part,we develop the entropy satisfying higher order DIRK local dis-continuous Galerkin methods for nonlinear degenerate parabolic equations.For some of these problems,it has been proved that the transient solutions converge to steady-states when time tends to infinity.With simple alternating numerical flux,we construct the DIRK local discontinuous Galerkin discretizations,which contain the advantages of higher order,entropy dissipation,steady-states preservation and long-time behavior capturing.The implicit time-discrete method greatly enlarge the time-step needed for stability of the numerical scheme.The larger time step and the simple alternating numer-ical flux significantly simplify the numerical computation.The entropy dissipation of the semi-discrete scheme and the entropy dissipation and steady-states preservation of the first order DIRK local discontinuous Galerkin discretization are deduced in theory.In order to ensure the positivity and the mass conservation of the numerical solution,we take the KKT limiter,which couples the positivity inequality constraint and the mass conservation equality constraint with higher order DIRK local discontinuous Galerkin discretizations using Lagrange multipliers.Numerical results are shown to illustrate the higher order accuracy and high efficiency of the positivity preserving DIRK local discontinuous Galerkin discretizations.