Related papers: A finite difference scheme for conservation laws d…
We present a fully discrete particle approximation for one-dimensional scalar conservation laws. Under suitable monotonicity assumptions on the macroscopic velocity, we construct a vacuum-compatible family of time-discrete particle…
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference…
This work is devoted to deriving small mass limiting equation for a class of Hamiltonian systems with multiplicative L\'evy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which…
The modified Hunter--Saxton equation models the propagation of short capillary-gravity waves. As it involves a mixed derivative, its initial value problem on the periodic domain is much more complicated than the standard evolutionary…
Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be…
In this paper, we establish a large deviation principle for a type of stochastic partial differential equations (SPDEs) with locally monotone coefficients driven by L\'evy noise. The weak convergence method plays an important role.
We develop a semi-discretization approximation for scalar conservation laws with multiple rough time dependence in inhomogeneous fluxes. The method is based on Brenier's transport-collapse algorithm and uses characteristics defined in the…
Scalar conservation laws sit at the intersection between being simple enough to study analytically, while being complex enough to exhibit a wide range of nonlinear phenomena. We introduce a novel stochastic perturbation of scalar…
This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes…
We analyze monotone difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L1…
We present a class of numerical schemes for two-dimensional systems of nonlocal conservation laws, which are based on utilizing well-known monotone numerical flux functions after suitably approximating the nonlocal terms. The considered…
We describe a methodology to build vectorial kinetic schemes, targetting the numerical solution of linear symmetric-hyperbolic systems of conservation laws -a minimal application case for those schemes. Precisely, we fully detail the…
We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are not necessarily unique in the…
We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on…
In this article we discuss the numerical analysis for the finite difference scheme of the one-dimensional nonlinear wave equations with dynamic boundary conditions. From the viewpoint of the discrete variational derivative method we propose…
In this paper we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic…
Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of…
In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by L\'{e}vy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and…
We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is…
We study the long-time behavior and the regularity of pathwise entropy solutions to stochastic scalar conservation laws with random in time spatially homogeneous fluxes and periodic initial data. We prove that the solutions converge to…