Unconditionally Stable Methods for Hamilton–Jacobi Equations

Karlsen, Kenneth H.; Risebro, Nils Henrik

doi:10.1006/jcph.2002.7113

Cited by 17 publications

(11 citation statements)

References 41 publications

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“…The purely convective version of (1.1) (A (u) ≡ 0) provides a simple model of traffic flow on a highway [49,25], the spatially varying coefficient γ corresponding to varying road conditions. Scalar conservation laws with discontinuities in the flux also arise in radar shape-from-shading problems [41] and as building blocks in dimensional splitting methods for multi-dimensional Hamilton-Jacobi equations [30]. Before continuing, let us detail the assumptions that we need to impose on the "data" of the problem (1.1).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

Karlsen

Risebro²,

Towers³

2002

IMA Journal of Numerical Analysis

107

153

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We analyze approximate solutions generated by an upwind difference scheme (of Engquist-Osher type) for nonlinear degenerate parabolic convection-diffusion equations where the nonlinear convective flux function has a discontinuous coefficient γ(x) and the diffusion function A(u) is allowed to be strongly degenerate (the pure hyperbolic case is included in our setup). The main problem is obtaining a uniform bound on the total variation of the difference approximation u Δ , which is a manifestation of resonance. To circumvent this analytical problem, we construct a singular mapping Ψ(γ, ·) such that the total variation of the transformed variable z Δ = Ψ(γ Δ , u Δ ) can be bounded uniformly in Δ. This establishes strong L 1 compactness of z Δ and, since Ψ(γ, ·) is invertible, also u Δ . Our singular mapping is novel in that it incorporates a contribution from the diffusion function A(u). We then show that the limit of a converging sequence of difference approximations is a weak solution as well as satisfying a Kružkov-type entropy inequality. We prove that the diffusion function A(u) is Hölder continuous, implying that the constructed weak solution u is continuous in those regions where the diffusion is nondegenerate. Finally, some numerical experiments are presented and discussed.1991 Mathematics Subject Classification. 35K65, 35L45, 35L65, 65M06, 65M12.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

Karlsen

Risebro²,

Towers³

2002

IMA Journal of Numerical Analysis

107

153

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“…There are a number of prototype relevant models with discontinuous nonlinearities in oil trapping phenomenon [9,10,23,50,70,78], a Whitham model of car traffic flow on a highway [49,67] and a model of continuous sedimentation in ideal clarifier-thickener units [21], see also [17,52]. Scalar equations and systems with discontinuous nonlinearities also arise in sedimentation processes [43], in radar shape-from-shading problems [72] and as well as building blocks in numerical methods for Hamilton-Jacobi equations [51]. Equations of the form (1) with discontinuous in x nonlinearity f attracted much attention in the recent past, because of the difficulties of adaptation of the approach developed for the smooth case, due in part to the presence of several different stable solutions with same initial data.…”

Section: Introductionmentioning

confidence: 99%

Weak asymptotic methods for scalar equations and systems

Abreu

Colombeau

Panov

2016

Journal of Mathematical Analysis and Applications

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Please cite this article in press as: E. Abreu et al., Weak asymptotic methods for scalar equations and systems, J. Math. Anal. Appl. (2016), http://dx. AbstractIn this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions. This construction produces weak asymptotic methods which are issued from Maslov asymptotic analysis. We obtain a sequence of functions which tend to satisfy the equation(s) in the weak sense in the space variable and in the strong sense in the time variable. To this end we reduce the partial differential equations to a family of ordinary differential equations in a classical Banach space. For scalar equations we prove that the initial value problem is well posed in the L 1 sense for the approximate solutions we construct. Then we prove that this method gives back the widely accepted solutions when they are known. For systems we obtain existence in the general case and uniqueness in the analytic case using an abstract Cauchy-Kovalevska theorem.

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“…We mention here briefly flow in porous media [15], sedimentation processes [5,13,14], and traffic flow on a highway [57,17]. They also arise in radar shape-from-shading problems [44] and as building blocks in numerical methods for Hamilton-Jacobi equations [21] based on dimensional splitting. In view of their applications, there is great demand for accurate, efficient, and, at the same time, easy-to-implement numerical methods for conservation laws with discontinuous coefficients.…”

Section: Introductionmentioning

confidence: 99%

An Engquist–Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections

Bürger¹,

Karlsen²,

Towers³

2009

SIAM J. Numer. Anal.

172

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Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV , we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat-Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver.

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Unconditionally Stable Methods for Hamilton–Jacobi Equations

Cited by 17 publications

References 41 publications

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

Weak asymptotic methods for scalar equations and systems

An Engquist–Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections

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