Rankine–Hugoniot–Riemann Solver Considering Source Terms and Multidimensional Effects

Jenny, Patrick; Müller, Bernhard

doi:10.1006/jcph.1998.6037

Cited by 30 publications

(29 citation statements)

References 22 publications

(33 reference statements)

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“…Such systems of equations can describe a number of physical phenomena, e.g. combustion [1], multiphase flow with phase interaction in the form of mass or heat transfer [2,3], water/vapour flow in nuclear reactors [4], cavitation [5], shallow water flow over variable topography [6,7], and fluid flow in a gravity field [8], to mention a few. In many cases, one can express the equations as balance laws consisting of a conservation law together with a source term, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…One common approach to solving equations of the form (1) is to use a fractional-step or operator-splitting method, which is based on solving the conservation law u t +∇·F(u) = 0 and the ordinary differential equation (ODE) u t = q(u, x) alternately to approximate the solution of the full problem (1). The advantage of such a splitting approach is that the operators can be approximated using well-proven methods developed for homogeneous conservation laws and for ODEs, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…This in turn leads to altered Riemann problems at the cell boundaries, which can be solved using a standard approximate Riemann solver with first or higher order reconstruction. Jenny and Müller [1] used a similar idea, but rather placed the source term at the cell boundary. Their work also introduced the concept for 2D problems of treating the flux gradients in the y direction as source terms when solving the Riemann problems in the x direction, and vice versa.…”

Section: Introductionmentioning

confidence: 99%

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Rankine–Hugoniot–Riemann solver for steady multidimensional conservation laws with source terms

Lund

Müller

et al. 2014

Computers & Fluids

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The Rankine-Hugoniot-Riemann (RHR) solver has been designed to solve steady multidimensional conservation laws with source terms. The solver uses a novel way of incorporating cross fluxes as source terms. The combined source term from the cross fluxes and normal source terms is imposed in the middle of a cell, causing a jump in the solution according to the Rankine-Hugoniot condition. The resulting Riemann problems at the cell faces are then solved by a conventional Riemann solver.We prove that the solver is of second order accuracy for rectangular grids and confirm this by its application to the 2D scalar advection equation, the 2D isothermal Euler equations and the 2D shallow water equations. For these cases, the error of the RHR solver is comparable to or smaller than that of a standard Riemann solver with a MUSCL scheme. The RHR solver is also applied to the 2D full Euler equations for a channel flow with injection, and shown to be comparable to a MUSCL solver.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rankine–Hugoniot–Riemann solver for steady multidimensional conservation laws with source terms

Lund

Müller

et al. 2014

Computers & Fluids

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“…The balance (1.4) is achieved in his quasi-steady method by introducing additional Riemann problems at the cell center of each grid cell whose flux difference cancel exactly the source term. A similar approach can be found in Jenny and Müller [41] and their so-called Rankine-Hugoniot Riemann solver. In Zhou, Causon, Mingham, and Ingram [73], the authors derive a well-balanced scheme for the shallow water equations with bottom topography, which is based on the Harten-Lax-van Leer (HLL) approximate Riemann solver and the surface gradient method.…”

Section: Introductionmentioning

confidence: 99%

Front Tracking for Scalar Balance Equations

Karlsen

Risebro

Towers

2004

J. Hyper. Differential Equations

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Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL-condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady-state solutions (or achieving them in the long time limit) with good accuracy.

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“…Many approaches have been studied for equations of this form, primarily in the case of an autonomous flux function f (q) (e.g., [4,8,11,13,14,28,30]). One simple approach that is often used is the fractional step method, in which one alternates between solving the homogeneous equation (1.1) and the ordinary differential equation…”

Section: Source Terms and Balance Laws The Balance Lawmentioning

confidence: 99%

A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions

Bale

LeVeque

2003

SIAM J. Sci. Comput.

239

295

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Abstract. We study a general approach to solving conservation laws of the form qt+f (q, x)x = 0, where the flux function f (q, x) has explicit spatial variation. Finite-volume methods are used in which the flux is discretized spatially, giving a function f i (q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A high-resolution wave-propagation algorithm is defined in which waves are based directly on a decomposition of flux differences f i (Q i ) − f i−1 (Q i−1 ) into eigenvectors of an approximate Jacobian matrix. This method is shown to be second-order accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws qt + f (q, x)x = ψ(q, x) are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasi-steady problems close to steady state.

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Rankine–Hugoniot–Riemann Solver Considering Source Terms and Multidimensional Effects

Cited by 30 publications

References 22 publications

Rankine–Hugoniot–Riemann solver for steady multidimensional conservation laws with source terms

Rankine–Hugoniot–Riemann solver for steady multidimensional conservation laws with source terms

Front Tracking for Scalar Balance Equations

A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions

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