In this paper, we consider the multi-symplectic Runge–Kutta (MSRK) methods applied to the nonlinear Dirac equation in relativistic quantum physics, based on a discovery of the multi-symplecticity of the equation. In particular, the conservation of energy, momentum and charge under MSRK discretizations is investigated by means of numerical experiments and numerical comparisons with non-MSRK meth...

This is an expository paper discussing the regularity and large time behavior of admissible BV solutions of genuinely nonlinear, strictly hyperbolic systems of two conservation laws. The approach will be via the theory of generalized characteristics.

We study the propagation of δ-shock wave in a new type of system of conservation laws. The particular cases of this system are the system of nonlinear chromatography and the system for isotachophoresis.

This paper studies the generalized chiral solitons. The integration of the generalized version of the chiral nonlinear Schrödinger's equation is obtained. The conservation laws are also computed using the multiplier approach. PACS Codes: 02.20.Sv; 02.30.Jr; 02.30.Ik.

We propose a fully conservative Front Tracking algorithm for systems of nonlinear conservation laws. The algorithm can be applied uniformly in one, two, three and N dimensions. Implementation details for this algorithm and tests of fully conservative simulations are reported.

We discuss some recent developments and ideas in studying the compactness and asymptotic behavior of entropy solutions without locally bounded variation for nonlinear hyperbolic systems of conservation laws. Several classes of nonlinear hyperbolic systems with resonant or linear degeneracy are analyzed. The relation of the asymptotic problems to other topics such as scale-invariance, compactnes...

We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniquenes...

In this paper we present the analysis for the Runge-Kutta discontinuous Galerkin (RKDG) method to solve scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge–Kutta (TVDRK3) method. We use an energy technique to present the L-norm stability for scalar linear conservation laws, and obtain a priori error estimates for smooth solutions...

Using inverse positivity properties and Brouwer’s fixed point theorem, we derive existence and uniqueness results for certain nonlinear systems of equations with off diagonal nonlinearity. The type of systems considered arises from stable finite volume discretizations of viscous nonlinear conservation laws and has a wide range of applications.

The solutions of the nonlinear Schrödinger equation show interesting behavior like recurring solutions and are of great importance for ab-initio calculations. It can be shown that its solutions conserve a countable number of quantities, the simplest being mass. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we...