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In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the L 2 L^2 norm of the numerical solution does not increase in time, under the time step condition τ ≤ F ( h / c , d / c 2 ) \tau \le \mathcal {F}(h/c, d/c^2) , with the convection coefficient c c , the diffusion coefficient d d , and the mesh size h h . The function F \mathcal {F} depends on the specific IMEX temporal method, the polynomial degree k k of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes τ ≲ h / c \tau \lesssim h/c in the convection-dominated regime and it becomes τ ≲ d / c 2 \tau \lesssim d/c^2 in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)-th order of accuracy when using piecewise k-th degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation.

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

We prove a cell entropy inequality for a class of high-order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one-dimensional scalar convex case.

In this paper we study the two-dimensional version of the Runge-Kutta Local Projection Discontinuous Galerkin (RKDG) methods, already defined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-ordrr accurate. Preliminary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.

We continue our investigation of overcoming the Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C k μ ( x ) C_k^\mu (x) with the weight function ( 1 − x 2 ) μ − 1 / 2 {(1 - {x^2})^{\mu - 1/2}} for any constant μ ≥ 0 \mu \geq 0 , of an L 1 {L_1} function f ( x ) f(x) , we can construct an exponentially convergent approximation to the point values of f ( x ) f(x) in any subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.

This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws u t + ∑ i = 1 d ( f i ( u ) ) x i = 0 {u_t} + \sum \nolimits _{i = 1}^d {{{({f_i}(u))}_{{x_i}}} = 0} . In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.

In the computation of conservation laws u t + f ( u ) x = 0 {u_t} + f{(u)_x} = 0 , TVD (total-variation-diminishing) schemes have been very successful. But there is a severe disadvantage of all TVD schemes: They must degenerate locally to first-order accuracy at nonsonic critical points. In this paper we describe a procedure to obtain TVB (total-variation-bounded) schemes which are of uniformly high-order accuracy in space including at critical points. Together with a TVD high-order time discretization (discussed in a separate paper), we may have globally high-order in space and time TVB schemes. Numerical examples are provided to illustrate these schemes.

In this paper, we present an essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.