Stability Analysis And Error Estimates Of The Fully Discrete Local Discontinuous Galerkin Method For Three Types Of Higher Order Evolution Equations

The local discontinuous Galerkin method has been widely used to solve higherorder partial differential equations since it was proposed,because of its advantage of stable numerical format and high accuracy.Hence it is particularly important to study the fully discrete method.We studied the stability and error estimates of the explicit and the implicit Runge-Kutta fully discrete local discontinuous Galerkin method for three types of higher order evolution equations,respectively,in this paper.In the second chapter,we presented the stability and the error estimate of the strong stability preserving implicit Runge-Kutta fully discrete local discontinuous Galerkin method for solving the linear bi-harmonic equation.According to the generalized alternating numerical fluxes we selected,the relationship between the numerical solution and the inner product of the auxiliary solution is established.By carefully introducing reference functions and using the generalized Gauss-Radau projection,we demonstrated the optimal convergence order of the fully discrete method for the linear bi-harmonic equation.Finally,we verified the optimal convergence conclusion through numerical experiments.In the third chapter,the stability and the error estimate of the fully discrete format with the arbitrary orders and arbitrary stages explicit Runge-Kutta scheme in time discretization and the local discontinuous Galerkin method in space for solving the linear third-order Korteweg-de Vries equation are studied.With the help of the temporal difference of numerical solution at each time stage,we established the energy equation and proved the stability of the fully discrete method by the aid of the matrix transferring process.By defining reference functions and using the general GaussRadau projection,we obtained the optimal convergence accuracy of order k +1.Finally,we calculated the linear and nonlinear Korteweg-de Vries equations respectively to verify the conclusion of convergence.In the fourth chapter,for the nonlinear fifth-order equation,we discussed the stability of the fully discrete method with the first order implicit Runge-Kutta scheme in time discretization and the ultra-weak local discontinuous Galerkin method in spatial discretization.The relationship between the auxiliary solution and the nonlinear term is established,also the stability of the first-order implicit Runge-Kutta fully discrete method is verified by analyzing the positiveness and negativity of the defined nonlinear operators.