Finite volume methods for irregular one-dimensional grids

Berger, Marsha; LeVeque, R; Stern, L.G.

doi:10.1090/psapm/048/1314852

Cited by 3 publications

(4 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [7], [28] h-box values were defined in a similar way, and the resulting scheme was shown to give good results for advection and Burgers's equation. One time step of the h-box method based on the interpolation formula (3.3) again for a > 0 has in the small cell the form…”

Section: H)mentioning

confidence: 99%

See 1 more Smart Citation

H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

Berger¹,

Helzel²,

LeVeque³

2003

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. We study generalizations of the high-resolution wave propagation algorithm for the approximation of hyperbolic conservation laws on irregular grids that have a time step restriction based on a reference grid cell length that can be orders of magnitude larger than the smallest grid cell arising in the discretization. This Godunov-type scheme calculates fluxes at cell interfaces by solving Riemann problems defined over boxes of a reference grid cell length h.We discuss stability and accuracy of the resulting so-called h-box methods for one-dimensional systems of conservation laws. An extension of the method for the two-dimensional case, which is based on the multidimensional wave propagation algorithm, is also described.

show abstract

Section: H)mentioning

confidence: 99%

“…An extension of h-box methods to a completely irregular grid was considered by Berger, LeVeque, and Stern [7]; see also Stern [28]. We will consider such calculations in section 5.…”

Section: Introductionmentioning

confidence: 99%

H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

Berger¹,

Helzel²,

LeVeque³

2003

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Although customary, such a condition might be unreasonably stringent for most of the elements. An interesting alternative is discussed in [1].…”

Section: The Numerical Schemes and The Main Resultsmentioning

confidence: 99%

A priori error estimates for numerical methods for scalar conservation laws. Part II : flux-splitting monotone schemes on irregular Cartesian grids

Cockburn

Gremaud

1997

Math. Comp.

View full text Add to dashboard Cite

Abstract. This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of (∆x) 1/2 in L ∞ (L 1 ) for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called p−consistent schemes, and prove that they converge to the entropy solution with the rate of (∆x) min{1/2,p} in L ∞ (L 1 ); again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of (∆x) 1/2 in L ∞ (L 1 ). We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

show abstract

“…Actually in Wendroff and White [31], [30] and Wendroff [29], a grid function is introduced for suitably smooth solutions of hyperbolic system in order to prove the optimal order of convergence of upwind finite difference and Lax-Wendroff schemes in one dimension space and in two dimension when an Alternate Direction Method is used. Berger et al [4], [3] successfully used the idea to get the optimal rate of convergence of the "h-box" scheme defined on a non-uniform cartesian grid with embedded irregular small cells. Recently Després [9], [8], by using energy estimates, has given an elegant proof of an order h 1/2 error estimates with respect to the L 2 norm for the linear advection equation on regular finite element meshes in the particular case of periodic boundary conditions.…”

mentioning

confidence: 99%

Error Estimate and the Geometric Corrector for the Upwind Finite Volume Method Applied to the Linear Advection Equation

Bouche¹,

Ghidaglia²,

Pascal³

2005

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in R nd and every non vanishing vector a of R nd. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs. Then we prove that this norm is indeed bounded by the mesh size in various cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimension. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases like the one with independent refined meshes.

show abstract

Finite volume methods for irregular one-dimensional grids

Cited by 3 publications

References 0 publications

H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

A priori error estimates for numerical methods for scalar conservation laws. Part II : flux-splitting monotone schemes on irregular Cartesian grids

Error Estimate and the Geometric Corrector for the Upwind Finite Volume Method Applied to the Linear Advection Equation

Contact Info

Product

Resources

About