The Riemann Problem in Two Space Dimensions for a Single Conservation Law

Wagner, David

doi:10.1137/0514045

Cited by 99 publications

(46 citation statements)

References 7 publications

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“…[8,25,27]. In 1986, the four-constant two-dimensional Riemann problem for a typical scalar conservation law was solved, see [34,37]. For the full Euler, a set of educated guesses of solutions to the four-wave two-dimensional Riemann problems was proposed in 1990 [38].…”

Section: Early Resultsmentioning

confidence: 99%

Two-dimensional Riemann problems for the compressible Euler system

Zheng

2009

Chin. Ann. Math. Ser. B

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Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. We list merits of one-dimensional Riemann problems and compare them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. We approach twodimensional Riemann problems via the methodology promoted by Andy Majda in the spirits of modern applied mathematics; that is, simplified model building via asymptotic analysis, numerical simulation, and theoretical analysis. We derive a simplified model, called the pressure gradient system, from the full Euler system via an asymptotic process. We use state-of-the-art numerical methods in numerical simulations to discern small-scale structures of the solutions, e.g., semi-hyperbolic patches. We use analytical methods to establish the validity of the structure revealed in the numerical simulation. The entire process, used in many of Majda's programs, is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws. SystemsWe consider the two-dimensional (2-D) compressible Euler systemwhere ρ is density, u is velocity vector, p is pressure, E = |u| 2 /2 + γp/ρ is the total energy density, and γ > 1 is the gas constant. We also consider the so-called pressure gradient system

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Section: Early Resultsmentioning

confidence: 99%

Two-dimensional Riemann problems for the compressible Euler system

Zheng

2009

Chin. Ann. Math. Ser. B

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“…Solutions of (0.4) and (0.5) when n = 2 and Uo is constant in each of the four quadrants were studied by D. Wagner in [9]. He assumed that f = (It, h) with g, f!f > 0 and It sufficiently close to h. Under these assumptions he constructed the solution and analyzed its qualitative properties.…”

Section: Jrnmentioning

confidence: 99%

Large-Time Behavior of Solutions to a Scalar Conservation Law in Several Space Dimensions

Bauman¹,

Phillips²

1986

Transactions of the American Mathematical Society

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“…In the one-dimensional case, the Riemann problem can be solved in closed form, and discontinuity curves in the (x, t) plane are simple straight lines passing through the origin. However, in two dimensions, only some special cases of the general Riemann problem have already been solved, and those display a very rich structure; see the works of Wagner [43], Lindquist [28,29], and Tong et al [41,42]. Thus, methods which allow triangulations that can be easily adapted to resolve this structure, should seriously be taken into consideration.…”

Section: Introductionmentioning

confidence: 99%

The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case

1990

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Abstract.In this paper we study the two-dimensional version of the RungeKutta Local Projection Discontinuous Galerkin (RKDG) methods, already defined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate. Preliminary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.

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The Riemann Problem in Two Space Dimensions for a Single Conservation Law

Cited by 99 publications

References 7 publications

Two-dimensional Riemann problems for the compressible Euler system

Two-dimensional Riemann problems for the compressible Euler system

Large-Time Behavior of Solutions to a Scalar Conservation Law in Several Space Dimensions

The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case

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