A semi-discrete Lagrangian–Eulerian scheme for hyperbolic-transport models

Abstract

In this work, we introduce a new semi-discrete scheme based on the so-called no-flow curves and its numerical analysis for solving initial value problems that involve one-dimensional (1D) scalar hyperbolic conservation laws and of the form, ut+H(u)x=0, x∈R,t>0, u(x,0)=u0(x). In addition, we present a two-dimensional (2D) version of the semi-discrete Lagrangian–Eulerian scheme to show that the proposed method can be applied to multidimensional problems. From an improved 1D weak numerical asymptotic analysis, we found that the solutions provided by the novel semi-discrete scheme satisfy a maximum principle property and a Kruzhkov entropy condition. We also highlight the possibility of using the no-flow curves as a new desingularization analysis technique for the construction of computationally stable numerical fluxes in the locally conservative form for nonlinear hyperbolic problems. We provide nontrivial 1D and 2D numerical examples with nonlinear wave interaction to illustrate the effectiveness and capabilities of the proposed approach and verify the theory. According to the results, the scheme handles discontinuous solutions (shocks) with low numerical dissipation quite well and shows a very good resolution of rarefaction waves with no spurious glitch effect in the vicinity of the sonic points. We also consider a test case for nonstrictly hyperbolic conservation laws with a resonance point (coincidence of eigenvalues) modeling three-phase flow in a porous media transport problem.