Stability analysis and error estimates of implicit Runge-Kutta local discontinuous Galerkin methods for linear bi-harmonic equation

In this paper, stability analysis and error estimates of a fully discrete local discontinuous Galerkin method for solving the linear bi-harmonic equation are carried out, where the time discretization is treated by a strong-stability-preserving implicit Runge-Kutta scheme. Based on the generalized alternating numerical fluxes, the relationship between the numerical solution and the inner product of the auxiliary solution is established, which plays a key role in obtaining the unconditional stability of the fully discrete local discontinuous Galerkin methods. By carefully introducing reference functions and generalized Gauss-Radau projection, the optimal error estimates are obtained. Numerical experiments are given to demonstrate the stability and accuracy of the fully discrete scheme for the linear bi-harmonic equation.