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SpaceTime Adaptive Solution of First Order PDEs
, 2003
"... An explicit timestepping method is developed for adaptive solution of timedependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a RungeKuttaFehlber ..."
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An explicit timestepping method is developed for adaptive solution of timedependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a RungeKuttaFehlberg method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. The error equation is integrated to obtain global errors of the solution. The method is shown to be stable if onesided space discretizations are used. Examples such as the wave equation, Burgers’ equation, and the Euler equations in one space dimension with discontinuous solutions illustrate the method.
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.
Entropy FluxSplittings For Hyperbolic Conservation Laws Part I: General Framework
 Comm. Pure Appl. Math
, 1995
"... A general framework is proposed for the derivation and analysis of fluxsplittings and the corresponding fluxsplitting schemes for systems of conservation laws endowed with a strictly convex entropy. The approach leads to several new properties of the existing fluxsplittings and to a method for the ..."
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A general framework is proposed for the derivation and analysis of fluxsplittings and the corresponding fluxsplitting schemes for systems of conservation laws endowed with a strictly convex entropy. The approach leads to several new properties of the existing fluxsplittings and to a method for the construction of entropy fluxsplittings for general situations. A large family of genuine entropy fluxsplittings is derived for several significant examples: the scalar conservation laws, the psystem, and the Euler system of isentropic gas dynamics. In particular, for the isentropic Euler system, we obtain a family of splittings that satisfy the entropy inequality associated with the mechanical energy. For this system, it is proved that there exists a unique genuine entropy fluxsplitting that satisfies all of the entropy inequalities, which is also the unique diagonalizable splitting. This splitting can be also derived by the socalled kinetic formulation. Simple and useful difference sc...
Adaptive mesh refinement using subdivision of Unstructured elements for conservation laws
, 2003
"... I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged. 1 ..."
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I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged. 1
Continuous kinematic wave models of merging traffic flow
, 810
"... Traffic dynamics at a merging junction can be numerically solved with discrete conservation equations and socalled supplydemand methods. In this paper, we first introduce a continuous multicommodity kinematic wave model of merging traffic and then develop a new framework for constructing the solu ..."
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Traffic dynamics at a merging junction can be numerically solved with discrete conservation equations and socalled supplydemand methods. In this paper, we first introduce a continuous multicommodity kinematic wave model of merging traffic and then develop a new framework for constructing the solutions to its Riemann problem with jump initial conditions. In the supplydemand space, the solutions on a link consist of an interior state and a stationary state, subject to admissible conditions such that there are no positive and negative kinematic waves on the upstream and downstream links respectively. In addition, the solutions have to satisfy entropy conditions defined by the supplydemand method in the interior states and a corresponding distribution scheme. For a merging junction with two upstream links, we prove that the stationary states and boundary fluxes exist and are unique for the Riemann problem for both fair and constant distribution schemes. With a numerical example, we demonstrate that the boundary fluxes converge to the analytical solutions at any positive time when we decrease the period of a time interval.
pressureinv[fiWKx conservfiWKx GodunovfiWKx method for barotropic twofluid flows
"... Discretizations of twofluid flow problems inconservx/fiV formulation generally exhibit pressure oscillations. In this work we show that these pressure oscillations are induced by the loss of apressureinvKvv property under discretization, and we introduce a nonoscillatoryconservllat method for bar ..."
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Discretizations of twofluid flow problems inconservx/fiV formulation generally exhibit pressure oscillations. In this work we show that these pressure oscillations are induced by the loss of apressureinvKvv property under discretization, and we introduce a nonoscillatoryconservllat method for barotropic twofluid flows. TheconservKx/T formulation renders the twofluid flow problem suitable to treatment by aGodunovfi[/S method. We present a modified Osher scheme for the twofluid flow problem. Numerical results are presented for a translatinginterface test case and a shock/interfacecollision test case.
New NAG Library Software Partial Differential Equations for FirstOrder
"... New NAG Fortran Library routines are described for the solution of systems of nonlinear, firstorder, timedependent partial differential equations in one space dimension, with scope for coupled ordinary differential or algebraic equations. The methodoflines is used with spatial discretization by ..."
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New NAG Fortran Library routines are described for the solution of systems of nonlinear, firstorder, timedependent partial differential equations in one space dimension, with scope for coupled ordinary differential or algebraic equations. The methodoflines is used with spatial discretization by either the centraldifference Keller box scheme or an upwind scheme for hyperbolic systems of conservation laws. The new routines have the same structure as existing library routines for the solution of secondorder partial differential equations, and much of the existing library software is reused. Results are presented for several computational examples to show that the software provides physically realistic numerical solutions to a challenging class of problems.
BLOCKSTRUCTURED ADAPTIVE MESH REFINEMENT THEORY, IMPLEMENTATION AND APPLICATION
"... Abstract. Structured adaptive mesh refinement (SAMR) techniques can enable cuttingedge simulations of problems governed by conservation laws. Focusing on the strictly hyperbolic case, these notes explain all algorithmic and mathematical details of a technically relevant implementation tailored for ..."
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Abstract. Structured adaptive mesh refinement (SAMR) techniques can enable cuttingedge simulations of problems governed by conservation laws. Focusing on the strictly hyperbolic case, these notes explain all algorithmic and mathematical details of a technically relevant implementation tailored for distributed memory computers. An overview of the background of commonly used finite volume discretizations for gas dynamics is included and typical benchmarks to quantify accuracy and performance of the dynamically adaptive code are discussed. Largescale simulations of shockinduced realistic combustion in nonCartesian geometry and shockdriven fluidstructure interaction with fully coupled dynamic boundary motion demonstrate the applicability of the discussed techniques for complex