Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws

Aregba-Driollet, Denise; Natalini, Roberto

doi:10.1137/s0036142998343075

Cited by 140 publications

(164 citation statements)

References 62 publications

Supporting

Mentioning

152

Contrasting

Order By: Relevance

“…This scheme is based on a relaxation scheme for the spatial discretization of the transport part [4]. We set the numerical velocity λ as the maximum of the eigenvalues of the Jacobian matrices of the fluxes A 1 and A 2 , namely, λ = max{2v L1 , v S1 − √ γ, v S1 + √ γ, 2v L2 , v S2 − √ γ, v S2 + √ γ}; space step is fixed, whereas the time step varies and satisfies the stability condition λ δt δ x ≤ 1.…”

Section: Numerical Schemementioning

confidence: 99%

“…A possible solution is to use moving fronts techniques (see [1]), which lead to other analytical and numerical difficulties and a further approximation in the model. Therefore, we keep the inertial terms and we solve the full hyperbolic problem using robust numerical schemes, as the Riemann Solver-free relaxation schemes [4]. There are two important differences with respect to a usual hyperbolic system.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A fluid dynamics model of the growth of phototrophic biofilms

et al. 2012

View full text Add to dashboard Cite

ABSTRACT. A system of nonlinear hyperbolic partial differential equations is derived using mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme will be described in detail and several simulations will be presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing. Fluid dynamics model and Hyperbolic equations and Phototrophic biofilms and Front propagation AMS : 92C17, 35L50, 65M06.

show abstract

Section: Numerical Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A fluid dynamics model of the growth of phototrophic biofilms

et al. 2012

View full text Add to dashboard Cite

show abstract

“…In [29] it was also shown how the above approach could be generalized to multidimensional systems of conservation laws in a natural way by adding further auxiliary variables and equations. Subsequently, the idea to approximate nonlinear PDEs by relaxation has been extended to diffusion and convection diffusion equations; see, for example, [1,2,28,29,32,36,37,11,13]. In particular, for nonlinear diffusion equations like (1.1), a relaxation system can be obtained introducing two auxiliary variables, as described in [36].…”

Section: A107mentioning

confidence: 99%

“…In Table 5.3, we compare the errors of the simulations at the same final time t = 2 for two different fixed numbers of spatial elements (N = 20 and N = 640) and for several spatial and temporal approximations with stabilization parameter α = h −1 (1,0). Also in this case, rotating α does not significantly affect the simulation results.…”

Section: Nonlinear Diffusion Testsmentioning

confidence: 99%

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Cavalli¹,

Naldi²,

Perugia³

2012

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ε. When ε → 0 + , the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing ε yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicitexplicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach. Key words. discontinuous Galerkin method, relaxation models, nonlinear diffusion AMS subject classifications. 65M60, 65M12, 35K55DOI. 10.1137/110827752 1. Introduction. Linear and nonlinear diffusion equations come from a variety of diffusion phenomena widely appearing in nature. They are suggested as mathematical models in many fields, such as filtration, phase transition, biochemistry, image analysis, and dynamics of biological groups. In the nonlinear case, the classical solutions to many of these PDEs fail to exist in finite time, even if the initial data are smooth. In such cases, suitable criteria have been introduced, which allow one to select physically relevant weak solutions beyond the singularity time.Recently, relaxation approximations to such PDEs have been introduced. These methods are based on replacing the equation by a semilinear hyperbolic system with stiff relaxation terms, tuned by a relaxation parameter ε. When ε → 0 + , the solution of this system "relaxes" onto the solution of the original PDE. Thus a consistent discretization of the relaxation system for ε = 0 yields a consistent discretization of the original PDE, as can be seen, for instance, in [29] and [2]. The advantage of this procedure is that the numerical scheme obtained in this fashion does not need approximate Riemann solvers for the convective term but possesses the robustness of upwind discretizations. Moreover, the complexity introduced by replacing the original

show abstract

“…However, special Runge-Kutta time stepping schemes have been proposed in [19,20,29] to develop high order relaxation schemes with MUSCL or WENO type space discretisations. It is interesting to note that the diagonal form of a Jin-Xin type relaxation system can be interpreted as a discrete velocity Boltzmann equation [1,4,33]. In the literature, lots of numerical studies have been reported in the context of both discrete Boltzmann and relaxation models, see [1,2,10,20,21,29,30,36,38,44] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

Arun

Lukáčová-Medviďová

Prasad

et al. 2013

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

View full text Add to dashboard Cite

Abstract. 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.

show abstract

Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws

Cited by 140 publications

References 62 publications

A fluid dynamics model of the growth of phototrophic biofilms

A fluid dynamics model of the growth of phototrophic biofilms

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

Contact Info

Product

Resources

About