Convergence of a Finite-Difference Scheme for Second-Order Hyperbolic Equations with Variable Coefficients

Jovanović, Boško S.; Ivanović, Lav; Süli, Endre

doi:10.1093/imanum/7.1.39

Cited by 16 publications

(17 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In 1986, Johnson and Pitkäranta [16] used a similar idea in the analysis of the DG method for linear hyperbolic problems. The question of post-processing the initial data is considered in the book of Brenner, Thomée and Wahlbin [6]; see also the work of Jovanović, Ivanović and Süli [17] concerning the use of convolution mollifiers with B-spline kernels for second-order hyperbolic boundary value problems with nonsmooth data.…”

Section: A Brief Overview Of the Development Of Post-processing Technmentioning

confidence: 99%

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

et al. 2002

View full text Add to dashboard Cite

Abstract. We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of ∆x only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k + 1/2 in the L 2 -norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L 2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L 2 (Ω 0 ), where Ω 0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

show abstract

Section: A Brief Overview Of the Development Of Post-processing Technmentioning

confidence: 99%

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

et al. 2002

View full text Add to dashboard Cite

show abstract

“…In the one-dimensional case, several authors studied the supraconvergence (see [8][9][10]15,24,32,37]). Also for hyperbolic and parabolic equations the supraconvergence was considered (see [2,18,29,39,42]). …”

Section: Introductionmentioning

confidence: 99%

Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

Emmrich

Grigorieff

2006

Comput. Methods Appl. Math.

View full text Add to dashboard Cite

-In this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. More precisely, for s ∈ (1/2, 2] the optimal order O(h s )-convergence of the finite difference solution and its gradient appears if the exact solution is in the Sobolev -Slobodetskij space H 1+s (Ω). All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose, i.e., the error of the approximate gradient with respect to the linear interpolation of the solution u is of the second order if u ∈ H 3 (Ω). A numerical example is given.2000 Mathematics Subject Classification: 65N06, 65N12, 65N15, 65N30.

show abstract

“…The current paper is therefore an attempt to design robust numerical approximations for the one-dimensional transport and the wave equation with rough, i.e., only Hölder continuous coefficients. For low enough Hölder exponent, this regularity requirement is less than the one in [11,12,10], and also our assumptions on the regularity of the solution are weaker. However, our results (so far) restrict to the one-dimensional case.…”

Section: Discussionmentioning

confidence: 96%

“…Thus, the design of numerical schemes that can approximate wave propagation in Hölder continuous media is a necessary first step in the efficient solution of the underlying uncertain PDE with a log-normal distributed material coefficient [20]. We are not aware of rigorous numerical analysis results for discretizations of the wave equation with such rough coefficients apart from the works [11,12,10] which require the coefficient to be in W s,2 (D) for some s ∈ (1, 3]. The current paper is therefore an attempt to design robust numerical approximations for the one-dimensional transport and the wave equation with rough, i.e., only Hölder continuous coefficients.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence rates of finite difference schemes for the linear advection and wave equation with rough coefficient

Weber

2016

IMA J Numer Anal

View full text Add to dashboard Cite

Abstract. We prove convergence rates of explicit finite difference schemes for the linear advection and wave equation in one space dimension with Hölder continuous coefficient. The obtained convergence rates explicitely depend on the Hölder regularity of the coefficient and the modulus of continuity of the initial data. We compare the theoretically established rates with the experimental rates of a couple of numerical examples.

show abstract

Convergence of a Finite-Difference Scheme for Second-Order Hyperbolic Equations with Variable Coefficients

Cited by 16 publications

References 0 publications

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

Convergence rates of finite difference schemes for the linear advection and wave equation with rough coefficient

Contact Info

Product

Resources

About