K. R. Arun

A WEAKLY ASYMPTOTIC PRESERVING LOW MACH NUMBER SCHEME FOR THE EULER EQUATIONS OF GAS DYNAMICS

We propose a low Mach number, Godunov-type finite volume scheme for the numerical solution of the compressible Euler equations of gas dynamics. The scheme combines Kleins non-stiff/stiff decomposition of the fluxes [J. Comput. Phys., 121 (1995), pp. 213--237] with an explicit/implicit time discretization [F. Cordier, P. Degond, and A. Kumbaro, J. Comput. Phys., 231 (2012), pp. 5685--5704] for the split fluxes. This results in a scalar second order partial differential equation (PDE) for the pressure, which we solve by an iterative approximation. Due to our choice of a crucial reference pressure, the stiff subsystem is hyperbolic, and the second order PDE for the pressure is elliptic. The scheme is also uniformly asymptotically consistent. Numerical experiments show that the scheme needs to be stabilized for low Mach numbers. Unfortunately, this affects the asymptotic consistency, which becomes nonuniform in the Mach number, and requires an unduly fine grid in the small Mach number limit. On the other han...

A Numerical Scheme for Three-Dimensional Front Propagation and Control of Jordan Mode

As an example of a front propagation, we study the propagation of a three-dimensional nonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions and geometry of the wavefront are obtained by solving the conservation form of equations of a weakly nonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of seven conservation laws with an additional vector constraint, each of whose components is a divergence-free condition. This constraint is an involution for the system of conservation laws, and it is termed a geometric solenoidal constraint. The analysis of a Cauchy problem for the linearized system shows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode. For the numerical simulation of the conservation laws we employ a high resolution central scheme. The second order accuracy of the scheme is achieved by using MUSCL-type reconstructions and Runge-Kutta time discretizations. A constrained transport-type technique is used to enforce the geometric solenoidal constraint. The results of several numerical experiments are presented, which confirm the efficiency and robustness of the proposed numerical method and the control of the Jordan mode.

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication K. R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R-3-in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293-311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 x 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accurate method is conservative, preserves the positivity of mass density and pressure and entropy stable. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. The results of numerical experiments on some benchmark problems confirm the efficiency and robustness of the proposed scheme.

A Characteristics Based Genuinely Multidimensional Discrete Kinetic Scheme for the Euler Equations

In this paper we present the results of a kinetic relaxation scheme for an arbitrary hyperbolic system of conservation laws in two space dimensions. We propose a new discrete velocity Boltzmann equation, which is an improvement over the previous models in terms of the isotropic coverage of the multidimensional domain by the foot of the characteristic. The discrete kinetic equation is solved by a splitting method consisting of a convection step and a collision step. The convection step involves only the solution of linear transport equations whereas the collision step instantaneously relaxes the distribution function to a local Maxwellian. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. Finally some numerical results are presented which confirm the robustness and correct multidimensional behaviour of the proposed scheme.

An application of 3-D kinematical conservation laws: propagation of a 3-D wavefront

Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface

Numerical Front Propagation Using Kinematical Conservation Laws

\Omega_t

IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows

in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic

Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions

7\times7

An application of 3-D Kinematical Conservation Laws: propagation of a three dimensional wavefront

system tha...