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Front tracking for scalar balance equations
 J. Hyperbolic Differ. Equ
"... 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 f ..."
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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 nonstiff 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. 1.
THE RIEMANN PROBLEM FOR THE SHALLOW WATER EQUATIONS WITH DISCONTINUOUS TOPOGRAPHY
, 2007
"... We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is nonstrictly hyperbolic and does not admit a fully conservative form, and we establish the existence of twoparameter wave sets, rather than wave curves ..."
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We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is nonstrictly hyperbolic and does not admit a fully conservative form, and we establish the existence of twoparameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit, and leads to formulas that can be implemented numerically for the approximation of the general initialvalue problem.
Fluxgradient and source term balancing for certain high resolution shockcapturing schemes
, 2006
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OPTIMAL CONTROL OF SYSTEMS OF CONSERVATION LAWS AND APPLICATION TO NONEQUILIBRIUM TRAFFIC STEERING
"... Abstract: This paper proposes an optimization algorithm to solve iteratively optimal control problems involving systems of conservation laws. The irregularity of their solution requires a specific variational analysis that have direct implications on the numerical implementation. This method is appl ..."
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Abstract: This paper proposes an optimization algorithm to solve iteratively optimal control problems involving systems of conservation laws. The irregularity of their solution requires a specific variational analysis that have direct implications on the numerical implementation. This method is applied to the control of nonequilibrium traffic using the PayneWhitham and the AwRascleZhang model.
Tsunami Modelling using the WellBalanced Scheme
"... Abstract—A well balanced numerical scheme based on stationary waves for shallow water flows with arbitrary topography has been introduced by Thanh et al. [18]. The scheme was constructed so that it maintains equilibrium states and tests indicate that it is stable and fast. Applying the wellbalanced ..."
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Abstract—A well balanced numerical scheme based on stationary waves for shallow water flows with arbitrary topography has been introduced by Thanh et al. [18]. The scheme was constructed so that it maintains equilibrium states and tests indicate that it is stable and fast. Applying the wellbalanced scheme for the onedimensional shallow water equations, we study the early shock waves propagation towards the Phuket coast in Southern Thailand during a hypothetical tsunami. The initial tsunami wave is generated in the deep ocean with the strength that of Indonesian tsunami of 2004. Keywords—Tsunami study, shallow water, conservation law, wellbalanced scheme, topography. Subject classification: 86 A 05, 86 A 17. I.
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"... A steadystate capturing method for hyperbolic systems with geometrical source terms ..."
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A steadystate capturing method for hyperbolic systems with geometrical source terms
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"... An efficient method for computing hyperbolic systems with geometrical source terms having concentrations ∗ ..."
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An efficient method for computing hyperbolic systems with geometrical source terms having concentrations ∗
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"... Two interface type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations ∗ ..."
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Two interface type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations ∗
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"... A numerical scheme for the quantum FokkerPlanckLandau equation efficient in the fluid regime ∗ ..."
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A numerical scheme for the quantum FokkerPlanckLandau equation efficient in the fluid regime ∗