Volume Schemes For Conservation Laws Research Articles

Semi-Conservative Finite Volume Schemes for Conservation Laws

This paper aims to introduce a new class of high order conservative schemes to solve systems of conservation laws. The idea is to couple the conservation form of the system with, possibly simpler, ...

Third‐ and fourth‐order well‐balanced schemes for the shallow water equations based on the CWENO reconstruction

SummaryHigh‐order finite volume schemes for conservation laws are very useful in applications, due to their ability to compute accurate solutions on quite coarse meshes and with very few restrictions on the kind of cells employed in the discretization. For balance laws, the ability to approximate up to machine precision relevant steady states allows the scheme to compute accurately, also on coarse meshes, small perturbations of such states, which are very relevant for many applications. In this paper, we propose third‐ and fourth‐order accurate finite volume schemes for the shallow water equations. The schemes have the well‐balanced property thanks to a path‐conservative approach applied to an appropriate nonconservative reformulation of the equations. High‐order accuracy is achieved by designing truly two‐dimensional (2D) reconstruction procedures of the central WENO (CWENO) type. The novel schemes are tested for accuracy and well‐balancing and shown to maintain positivity of the water height on wet/dry transitions. Finally, they are applied to simulate the Tohoku 2011 tsunami event.

Convergence of a flux-splitting finite volume scheme for conservation laws driven by Lévy noise

We explore numerical approximation of multidimensional stochastic balance laws driven by multiplicative Lévy noise via flux- splitting finite volume method. The convergence of the approximations is proved towards the unique entropy solution of the underlying problem.

Geometric error of finite volume schemes for conservation laws on evolving surfaces

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of $\mathbb{R}^3$. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.

Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

SUMMARYA complete set of equivalence conditions, relating the mass‐lumped Bubnov–Galerkin finite element (FE) scheme for Lagrangian linear elements to node‐centred finite volume (FV) schemes, is derived for the first time for conservation laws in a three‐dimensional cylindrical reference. Equivalence conditions are used to devise a class of FV schemes, in which all grid‐dependent quantities are defined in terms of FE integrals. Moreover, all relevant differential operators in the FV framework are consistent with their FE counterparts, thus opening the way to the development of consistent hybrid FV/FE schemes for conservation laws in a three‐dimensional cylindrical coordinate system. The two‐dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems and the interaction of a converging shock waves with obstacles, are carried out using the new approach. The results agree fairly well with one‐dimensional simulations and simplified models. Copyright © 2013 John Wiley & Sons, Ltd.

Vertex-centroid finite volume scheme on tetrahedral grids for conservation laws

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of both centroid and vertex values to construct high-resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure while the centroid values are updated by the finite volume scheme. A modified interpolation scheme is proposed which is better than existing schemes in giving positive weights in the interpolation formula. A simplified reconstruction scheme is also proposed which is also more efficient and leads to more robust schemes for discontinuous problems. For scalar conservation laws, we develop limited versions of the schemes which are stable in maximum norm by constructing suitable limiters. The schemes are applied to compressible flows governed by the Euler equations of inviscid gas dynamics.

Space–time adaptive multiresolution methods for hyperbolic conservation laws: Applications to compressible Euler equations

Adaptive strategies in space and time allow considerable speed-up of finite volume schemes for conservation laws, while controlling the accuracy of the discretization. In this paper, a multiresolution technique for finite volume schemes with explicit time discretization is presented. An adaptive grid is introduced by suitable thresholding of the wavelet coefficients, which maintains the accuracy of the finite volume scheme of the regular grid. Further speed-up is obtained by local scale-dependent time stepping, i.e., on large scales larger time steps can be used without violating the stability condition of the explicit scheme. Furthermore, an estimation of the truncation error in time, using embedded Runge–Kutta type schemes, guarantees a control of the time step for a given precision. The accuracy and efficiency of the fully adaptive method is illustrated with applications for compressible Euler equations in one and two space dimensions.

Fully Adaptive Multiscale Schemes for Conservation Laws Employing Locally Varying Time Stepping

In recent years the concept of fully adaptive multiscale finite volume schemes for conservation laws has been developed and analytically investigated. Here the grid adaptation is performed by means...

Fully adaptive multiresolution finite volume schemes for conservation laws

The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.

Finite Volume Schemes for Conservation Laws of Mixed Type

Systems of conservations laws are considered which can be viewed as coupled systems of elliptic and hyperbolic equations. For such systems a new concept of a one-dimensional Riemann problem is introduced. Thus a new Godunov-type scheme can be built in both the one-dimensional and the multidimensional cases.