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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.
SecondOrder Characteristic Methods for AdvectionDiffusion Equations and Comparison to Other Schemes
 Advances in Water Resources
, 1999
"... We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order RungeKutta approximation of the characteristics within the framework of the EulerianLagrangian localized adjoint method. These methods naturally incorporate all three type ..."
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We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order RungeKutta approximation of the characteristics within the framework of the EulerianLagrangian localized adjoint method. These methods naturally incorporate all three types of boundary conditions in their formulations, are fully mass conservative, and generate regularly structured systems which are symmetric and positive definite for most combinations of the boundary conditions. Extensive numerical experiments are presented which compare the performance of these two RungeKutta methods to many other well perceived and widely used methods which include many Galerkin methods and high resolution methods from #uid dynamics. Key words characteristic methods, comparison of numerical methods, EulerianLagrangian methods, numerical solutions of advectiondi#usion equations, RungeKutta methods. 1 Introduction Advectiondi#usion equations are an important cla...
UNIFIED METHODS FOR COMPUTING COMPRESSIBLE AND INCOMPRESSIBLE FLOWS
, 2000
"... To develop unified computing methods that are accurate and efficient both for compressible and incompressible flows, one may modify methods developed for the fully compressible case, or, viceversa, modify incompressible methods. Both approaches are reviewed. One leads to colocated, the other to st ..."
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To develop unified computing methods that are accurate and efficient both for compressible and incompressible flows, one may modify methods developed for the fully compressible case, or, viceversa, modify incompressible methods. Both approaches are reviewed. One leads to colocated, the other to staggered schemes. The latter resemble closely classical schemes for the hyperbolic systems of equations governing atmospheric and oceanographic flows. General equations of state are considered, leading to nonconvex hyperbolic systems. A simple way to solve a class of Riemann problems is presented, using Oleinik’s formulation of the entropy condition. The Osher scheme and a staggered scheme are found to converge with the same accuracy to the physical weak solution. The staggered scheme turns out to be useful for computing with the homogeneous equilibrium model a hydrodynamic flow with cavitation, in which a region where the Mach number reaches 25 is embedded in a domain where the Mach number equals 10 −3.
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.
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.
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
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...
Developments in Cartesian cut cell methods
"... This paper describes the Cartesian cut cell method, which provides a flexible and efficient alternative to traditional boundary fitted grid methods. The Cartesian cut cell approach uses a background Cartesian grid for the majority of the flow domain with special treatments being applied to cells whi ..."
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This paper describes the Cartesian cut cell method, which provides a flexible and efficient alternative to traditional boundary fitted grid methods. The Cartesian cut cell approach uses a background Cartesian grid for the majority of the flow domain with special treatments being applied to cells which are cut by solid bodies, thus retaining a boundary conforming grid. The development of the method is described with applications to problems involving both moving bodies and moving material interfaces.
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