Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions
Introduction
Hyperbolic conservation laws with spatially varying fluxes arise in many practical applications. For example, in modelling of acoustic, electromagnetics or elastic waves in heterogeneous materials or in the traffic flow with varying conditions. In exploration seismology one studies the propagation of small amplitude of man made waves in earth and their reflection off geological structures. For numerical modelling of wave propagation in heterogeneous media the reader is referred, for example, to [1], [8], [9], [10], [14], [23] and the references therein. A large variety of finite difference schemes for wave propagation can be found in particular in seismological literature; see, e.g., [2], [3], [7], [12], [24], [33], [34] just to mention some of them.
Our aim in this paper is to develop a new genuinely multi-dimensional method for approximation of hyperbolic conservation laws with spatially varying fluxes using the so-called evolution Galerkin framework. In particular, we will illustrate the methodology for the wave equation system with spatially varying wave speeds and simulate the propagation of acoustic waves in heterogeneous media. In our future study we would like to generalize ideas presented here to other models, for example for linear elastic waves.
The evolution (or characteristic) Galerkin schemes were first derived by Morton, Süli and their collaborators for scalar problems and for one-dimensional systems, see [15], [16] and the references therein. This research was motivated by the pioneering work of Butler [4] and the related works of Prasad et al. [31], [32]. In 2000 Lukáčová-Medvid’ová, Morton and Warnecke derived the Evolution Galerkin schemes for the linear wave equation system with constant wave speed [17]. In the recent works of Lukáčová-Medvid’ová et al. [11], [18], [19], [20] a genuinely multi-dimensional finite volume evolution Galerkin (FVEG) method has been developed. The FVEG scheme can be viewed as a predictor–corrector method; in the predictor step the data are evolved along the bicharacteristics to determine the approximate solution at cell interfaces. In the corrector step the finite volume update in conservative variables is realized. The method works well for linear as well as nonlinear hyperbolic systems. In order to derive evolution operators for nonlinear systems a suitable local linearization has been used. For a locally linearized system bicharacteristics are reduced to straight lines.
The goal of this paper is to derive the FVEG scheme for linear hyperbolic systems with spatially varying flux functions without any local linearization. In this case the Jacobians are spatially varying but time independent and bicharacteristics are no longer straight lines. This introduces new difficulties in the derivation of the exact integral representation as well as in the numerical approximation. In particular, we consider the acoustic wave equation system with a variable wave speed. The results presented here can be generalized to more complex hyperbolic conservation laws. However, we should note that an important property of our model is the fixed number of positive eigenvalues; indeed, as we will see in Section 2 eigenvalues do not pass through zero. Consequently, we are not facing the difficulties with development of delta functions as it might happen in a general case.
A mathematical model for propagation of acoustic waves can be derived from the conservative form of the Euler equations. One considers small perturbations of the background steady state , where denote respectively the density, -components of velocity and pressure. For simplicity, we assume that the gas is at rest initially, i.e. . It turns out from momentum equations of the Euler equations that has to be a constant. The acoustic waves are then governed by the following first order system, cf., e.g., [8]:Equivalently we havewhereand denotes the wave speed. We use (1.2) as our starting point.
In differential form this readswhere . Note that and . We develop the FVEG method for the system of conservation laws (1.4) in which the flux functions are non-constant functions of x and y.
The paper is organized as follows: in Section 2 we start with a brief review of characteristic theory in multi-dimensions to define the bicharacteristics of the wave equation system (1.3) and derive the exact integral representation along the bicharacteristics. In Section 3 the exact integral equations are approximated by numerical quadratures and suitable approximate evolution operators are derived. In Section 4 the first and second order finite volume evolution Galerkin scheme are constructed. We will show that it is preferable to model the heterogeneous medium by means of a staggered grid. In fact we approximate the wave speed and the impedance on a staggered grid. Finally, in Section 5 we illustrate the behaviour of the presented scheme on a set of numerical experiments for wave equation system with continuous as well as discontinuous wave speeds.
Section snippets
Bicharacteristics and exact integral representation
A characteristic surface of (1.3) is a possible surface of discontinuity in the first order derivatives of . The evolution of the surface is given by the eikonal equationwhere I is the identity matrix. Note that (2.1) is a scalar differential equation for . The characteristic curves of (2.1) are called the bicharacteristic curves of (1.3). These are curves in the space and can be obtained by solving the Charpit’s equations, cf.
Approximate evolution operator
In this section we approximate the exact integral representation (2.11), (2.15), (2.16) by suitable numerical quadratures and derive the corresponding approximate evolution operators.
Note that the exact integral equations contain time integrals involving the derivatives of the unknown variables. These are the terms that need our attention. First, let us consider in (2.15), (2.16) the integrals of and along a time like bicharacteristic. In order to eliminate these integrals we use the
Finite volume evolution Galerkin method
Let us divide a computational domain into a finite number of regular finite volumes for are the mesh steps in x- and y-directions, respectively. Denote by the piecewise constant approximate solution on a mesh cell at time and start with initial approximations obtained by the integral averages . Integrating the conservation law (1.2) and applying the Gauss theorem on any mesh cell yield the following
Numerical experiments
In this chapter we illustrate behaviour of the new FVEG method on a set of one- and two-dimensional experiments with continuous as well as discontinuous wave speeds. All experiments have been done with two-dimensional FVEG method. In the case of one-dimensional experiments we have imposed zero velocity and use simply the midpoint rule for the flux integration along cell interfaces. In all our experiments we have set the CFL number which is in agreement with our previous theoretical
Conclusions
In this paper we have generalized the genuinely multi-dimensional finite volume evolution Galerkin (FVEG) scheme to hyperbolic conservation laws with spatially varying flux functions. The methodology has been presented for acoustic waves in heterogeneous medium. The FVEG scheme is based on multi-dimensional approximate evolution operator that is used for prediction of fluxes on cell interfaces. Using general theory of bicharacteristics we have derived in the Sections 2 Bicharacteristics and
Acknowledgments
The present research has been supported under the DST-DAAD project based personnel exchange programme “Theory and numerics of multi-dimensional hyperbolic conservation laws and balance laws based on the bicharacteristics”. The authors gratefully acknowledge this support. K.R. Arun would like to express his gratitude to the Council of Scientific and Industrial Research (CSIR) for supporting his research at the Indian Institute of Science under the Grant 09/079(2084)/2006-EMR-1. The department of
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