5 research outputs found
Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems
We propose energy-conserving discontinuous Galerkin (DG) methods for
symmetric linear hyperbolic systems on general unstructured meshes. Optimal a
priori error estimates of order are obtained for the semi-discrete scheme
in one dimension, and in multi-dimensions on Cartesian meshes when
tensor-product polynomials of degree are used. A high-order
energy-conserving Lax-Wendroff time discretization is also presented.
Extensive numerical results in one dimension, and two dimensions on both
rectangular and triangular meshes are presented to support the theoretical
findings and to assess the new methods. One particular method (with the
doubling of unknowns) is found to be optimally convergent on triangular meshes
for all the examples considered in this paper. The method is also compared with
the classical (dissipative) upwinding DG method and (conservative) DG method
with a central flux. It is numerically observed for the new method to have a
superior performance for long-time simulations.Comment: 48 page
An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg-De Vries equation
We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method
for the generalized Korteweg-De Vries(KdV) equation in one dimension. Optimal a
priori error estimate of order is obtained for the semi-discrete scheme
for the KdV equation without convection term on general nonuniform meshes when
polynomials of degree is used. We also numerically observed optimal
convergence of the method for the KdV equation with linear or nonlinear
convection terms.
It is numerically observed for the new method to have a superior performance
for long-time simulations over existing DG methods.Comment: 12 pages. arXiv admin note: substantial text overlap with
arXiv:1804.1030
Dispersive behavior of an energy-conserving discontinuous Galerkin method for the one-way wave equation
The dispersive behavior of the recently proposed energy-conserving
discontinuous Galerkin (DG) method by Fu and Shu [10] is analyzed and compared
with the classical centered and upwinding DG schemes. It is shown that the new
scheme gives a significant improvement over the classical centered and
upwinding DG schemes in terms of dispersion error. Numerical results are
presented to support the theoretical findings.Comment: 17 pages, 6 figure
Strong stability of explicit Runge-Kutta time discretizations
Motivated by studies on fully discrete numerical schemes for linear
hyperbolic conservation laws, we present a framework on analyzing the strong
stability of explicit Runge-Kutta (RK) time discretizations for semi-negative
autonomous linear systems. The analysis is based on the energy method and can
be performed with the aid of a computer. Strong stability of various RK
methods, including a sixteen-stage embedded pair of order nine and eight, has
been examined under this framework. Based on numerous numerical observations,
we further characterize the features of strongly stable schemes. A both
necessary and sufficient condition is given for the strong stability of RK
methods of odd linear order
Error analysis of Runge--Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations
In this paper, we present error estimates of fully discrete Runge--Kutta
discontinuous Galerkin (DG) schemes for linear time-dependent partial
differential equations. The analysis applies to explicit Runge--Kutta time
discretizations of any order. For spatial discretization, a general discrete
operator is considered, which covers various DG methods, such as the
upwind-biased DG method, the central DG method, the local DG method and the
ultra-weak DG method. We obtain error estimates for stable and consistent fully
discrete schemes, if the solution is sufficiently smooth and a spatial operator
with certain properties exists. Applications to schemes for hyperbolic
conservation laws, the heat equation, the dispersive equation and the wave
equation are discussed. In particular, we provide an alternative proof of
optimal error estimates of local DG methods for equations with high order
derivatives in one dimension, which does not rely on energy inequalities of
auxiliary unknowns