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In this paper, we derive a new second-order fully discrete Godunov-type central- upwind scheme for two-dimensional hyperbolic systems of conservation laws. The scheme is derived in three steps: reconstruction, evolution, and projection. The novelty of our approach is in the evolution step, which is performed using the nonuniform quadrilateral control volumes obtained based on the one-sided local speeds of propagation, and in the projection step, in which the evolved solution is projected back onto the uniform grid with the help of a new sharp piecewise polynomial reconstruction. The scheme is tested on a number of numerical examples for the Euler equations of gas dynamics. We have demonstrated that the new scheme is nonoscillatory and at the same time it achieves higher resolution than the second-order semidiscrete central-upwind scheme. The latter suggests that the fully discrete scheme has a smaller amount of numerical dissipation. [ABSTRACT FROM AUTHOR]

Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Of particular interest here is the transport of species and energy in low Mach number models for combustion, atmospheric flows, and astrophysics, as well as contaminant transport in Darcy models of saturated subsurface flow. An important characteristic of these problems is that the velocity field is not known analytically. Instead, an auxiliary equation is solved to compute averages of the velocities over faces in a finite volume discretization. In this paper, we present a customized three-dimensional finite volume advection scheme for this class of problems that provides accurate resolution for smooth problems while avoiding undershoot and overshoot for nonsmooth profiles. The method is an extension of an algorithm by Bell, Dawson, and Shubin (BDS), which was developed for a class of scalar conservation laws arising in porous media flows in two dimensions. The original BDS algorithm is a variant of unsplit. higher-order Godunov methods based on construction of a limited bilinear profile within each computational cell. Here we present a three-dimensional extension of the original BDS algorithm that is based on a limited trilinear profile within each cell. We compare this new method to several other unsplit approaches, including piecewise linear methods, piecewise parabolic methods, and wave propagation schemes. [ABSTRACT FROM AUTHOR]

This paper presents a conservative front-tracking method for shocks and contact discontinuities that is second-order accurate. It is based on a volume-of-fluid method that treats the moving front with concepts similar to those of an embedded-boundary method. Special care is taken in the centering of the data to ensure the right order of accuracy at the front, and a redistribution step guarantees conservation. A suite of test problems, for tracking both shocks and contact discontinuities, is presented that confirms that the method is second-order accurate. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of a weakly imposed boundary condition for the one-dimensional Euler equations of gasdynamics. We give a mathematical proof of the convergence in two specific cases, and we analyze two kinds of physically relevant boundary conditions including the nonpenetration condition and the imposed pressure condition. We also show the results of a couple of one-dimensional test cases that confirm our theoretical estimates. [ABSTRACT FROM AUTHOR]

Hysteresis effects in two-phase flow in porous media are important in applications such as waterflooding or gas storage in sand aquifers. In this paper, we develop a numerical scheme for such a flow where the permeability hysteresis is modeled by a family of reversible scanning curves enclosed by irreversible imbibition and drainage permeability curves. The scheme is based on associated local Riemann solutions and can be viewed as a modification of the classical Godunov method. The Riemann solutions necessary for the scheme are presented, as well as the criteria that guarantee the well-posedness of the Riemann problem with respect to perturbations of left and right states. The numerical and analytical results show strong influence of the permeability hysteresis on the flow. In addition, the numerical scheme accurately reproduces the available experimental data once hysteresis is taken into account in the model. [ABSTRACT FROM AUTHOR]