An efficient and conservative hybrid method for solving multidimensional conservation laws

Abstract

We have developed an efficient hybrid technique for solving nonlinear conservation laws. Previous hybrid techniques have been accurate but lacked the property of conservation, whereas our technique is both accurate and conservative. To achieve this, we superimposed all possible stencils for ENO polynomials and weighted the value from each cells in a way that depends on the numerical solution. Computational efficiency relies on switching from central data where the exact solution is smooth to noncentral data near discontinuities. We prove the theoretical consistency of the technique and discuss the connection with ENO and WENO methods. We introduce time dependency by combining our method with Runge-Kutta schemes that are TVD preserving. We have verified our technique experimentally by solving a suite of test problems with convex and non-convex flux functions taken from the literature.

Finite-difference stable stencils based on least-square quadric fitting

“…In particular, good results have been obtained in the solution of convection problems and for the simulation of shock phenomena. Recent advances on this subject were achieved by hybrid methods which satisfy both accuracy and conservation laws, see [10]. Our work differs from the aforementioned studies by presenting a stable stencil for solving an ill posed problem during a short step, i.e.…”

“…solving the Laplace equation backwards. Special attention is placed on the discretization of the Laplace operator, while the work [10] is based on the discretization of the advection term.…”

A mass-conservative centered finite volume model for solving two-dimensional two-layer shallow water equations for fluid mud propagation over varying topography and dry areas

“…Requirements (30) and (31) provide four constraints on the function qi ðx; t n Þ for any triangular cell; here a least-square fitting is used to compute the slopes a i and b i [39][40][41].…”

Numerical Simulations of the Oscillating Second-Grade Fluid through a Rectangular Cross Duct with Fractional Constitution Relationship

“…There are many methods to solve the governing equation [29][30][31]. For the treatment of the fractional second-grade fluid, the traditional method is to apply the integral transform method to obtain an analytical solution [27,32,33], with the paradox that the principle of causality causing the initial conditions is a non-rigorous enforcement.…”