Abstract. The solutions of the nonlinear Schrödinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest being the local norm square conservation law. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we first giv...

We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [7, 6, 5, 4] for conservation Laws. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. We use this new formulation to solve conservation laws on one-d...

In this paper, we develop a new kind of multisymplectic integrator for the coupled nonlinear Schrödinger (CNLS) equations. The CNLS equations are cast into multisymplectic formulation. Then it is split into a linear multisymplectic formulation and a nonlinear Hamiltonian system. The space of the linear subproblem is approximated by a highorder compact (HOC) method which is new in multisymplecti...

In this paper, we investigate the accuracy-enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L2 norm. Therefore, we first prove that the L2 norm of...

Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, meteorology, electromagnetics, semi-conductor device simulation, models of biological processes and many other engineering areas governed by conservative systems that can be writt...

Any PDE system can be effectively analyzed through consideration of its tree of nonlocally related systems. If a given PDE system has n local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these n conservation laws, one can directly construct 2 − 1 independent nonlocally related systems by consider...

Any partial differential equation PDE system can be effectively analyzed through consideration of its tree of nonlocally related systems. If a given PDE system has n local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these n conservation laws, one can directly construct 2n−1 independent nonlocall...

The hierarchy of the integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of the linearization of these equations and their conservation law in the terms of the solutions of the corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of the lineari...

A smoothness/shock indicator is proposed for the RKDG methods solving nonlinear conservation laws. A few numerical experiments are presented as evidence that the indicator helps in detecting shocks, high order discontinuities, regions of smooth solutions, and numerical “instability”. keywords. Conservation law, discontinuous Galerkin method, smoothness indicator. AMS subject class. Primary: 65M...

ADER schemes are recent finite volume methods for hyperbolic conservation laws, which can be viewed as generalizations of the classical first order Godunov method to arbitrary high orders. In the ADER approach, high order polynomial reconstruction from cell averages is combined with high order flux evaluation, where the latter is done by solving generalized Riemann problems across cell interfac...