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38
APPROXIMATION OF HYPERBOLIC MODELS FOR CHEMOSENSITIVE MOVEMENT
, 2005
"... Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first and secondorder wellbalanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On th ..."
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Cited by 9 (2 self)
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Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first and secondorder wellbalanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On the other hand, a highorder finite difference weighted essentially nonoscillatory (WENO) scheme is constructed and the wellbalanced reconstruction is adapted to this scheme to exactly preserve steady states and to retain highorder accuracy. Numerical simulations are performed to verify accuracy and the wellbalanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments.
An asymptotic high order masspreserving scheme for a hyperbolic model of chemotaxis
 SIAM J. Num. Anal
"... Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutio ..."
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Cited by 8 (2 self)
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Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to non constant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. A special care is needed to deal with boundary conditions, to avoid harmful loss of mass. Convergence results are proven for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior.
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
"... ..."
How to solve systems of conservation laws numerically using the graphics processor as a highperformance computational engine
 Quak (Eds.), Geometric Modelling, Numerical Simulation, and Optimization: Industrial Mathematics at SINTEF
, 2005
"... Summary. The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax–Friedrichs scheme and the Lax–Wendroff scheme. Using a simple example, we show how the ..."
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Cited by 5 (2 self)
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Summary. The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax–Friedrichs scheme and the Lax–Wendroff scheme. Using a simple example, we show how these two schemes fail to give accurate approximations to solutions containing discontinuities. We then introduce a general class of semidiscrete finitevolume schemes that are designed to produce accurate resolution of both smooth and nonsmooth parts of the solution. Using this special class we wish to introduce the reader to the basic principles used to design modern highresolution schemes. As examples of systems of conservation laws, we consider the shallowwater equations for water waves and the Euler equations for the dynamics of an ideal gas. The second theme in the paper is how programmable graphics processor units (GPUs or graphics cards) can be used to efficiently compute numerical solutions of these systems. In contrast to instruction driven microprocessors (CPUs), GPUs subscribe to the datastreambased computing paradigm and have been optimised for high throughput of large data streams. Most modern numerical methods for hyperbolic conservation laws are explicit schemes defined over a grid, in which the unknowns at each grid point or in each grid cell can be updated independently of the others. Therefore such methods are particularly attractive for implementation using datastreambased processing. 1
A SUBSONICWELLBALANCED RECONSTRUCTION SCHEME FOR SHALLOW WATER FLOWS
"... Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reco ..."
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Cited by 5 (3 self)
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Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reconstruction that allows to derive a subsonicwellbalanced scheme, preserving exactly all the subsonic steady states. It generalizes the now wellknown hydrostatic solver, and as the latter it preserves nonnegativity of water height and satisfies a semidiscrete entropy inequality. An application to the EulerPoisson system is proposed. 1.
High Resolution Methods and Adaptive Refinement for Tsunami Propagation and Inundation.
"... We describe the extension of high resolution finite volume methods and adaptive refinement for the shallow water equations in the context of tsunami modeling. Godunovtype methods have been used extensively for modeling the shallow water equations in many contexts, however, tsunami modeling presents ..."
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Cited by 4 (1 self)
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We describe the extension of high resolution finite volume methods and adaptive refinement for the shallow water equations in the context of tsunami modeling. Godunovtype methods have been used extensively for modeling the shallow water equations in many contexts, however, tsunami modeling presents some unique challenges that must be overcome. We describe some of the specific difficulties associated with tsunami modeling, and summarize some numerical approaches that we have used to overcome these challenges. For instance, we have developed a wellbalanced Riemann solver that is appropriate in the deep ocean regime as well as robust in nearshore and dry regions. Additionally, we have extended adaptive refinement algorithms to this application. We briefly describe some of the modifications necessary for using these adaptive methods for tsunami modeling.
Upwinding of source term at interface for Euler equations with high friction, in "Computers and Mathematics with Applications
, 2006
"... with high friction ..."
Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation
 J. Comput. Phys
, 2008
"... We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Ty ..."
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Cited by 3 (1 self)
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We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Typically, a simple solver for a system of m conservation laws uses m such discontinuities. We present a four wave solver for use with the the shallow water equations—a system of two equations in one dimension. The solver is based on a decomposition of an augmented solution vector—the depth, momentum as well as momentum flux and bottom surface. By decomposing these four variables into four waves the solver is endowed with several desirable properties simultaneously. This solver is wellbalanced: it maintains a large class of steady states by the use of a properly defined steady state wave—a stationary jump discontinuity in the Riemann solution that acts as a source term. The form of this wave is introduced and described in detail. The solver also maintains depth nonnegativity and extends naturally to Riemann problems with an initial dry state. These are important properties for applications with steady states and inundation, such as tsunami and flood modeling. Implementing the solver with LeVeque’s wave propagation algorithm [25] is also described. Several numerical simulations are shown, including a test problem for tsunami modeling. Key words: shallow water equations, hyperbolic conservation laws, finite volume methods, Godunov methods, Riemann solvers, wave propagation, shock capturing methods, tsunami modeling
Quadtreeadaptive tsunami modelling
 Ocean Dynamics
, 2011
"... The wellbalanced, positivitypreserving scheme of Audusse et al, 2004, for the solution of the SaintVenant equations with wetting and drying, is generalised to an adaptive quadtree spatial discretisation. The scheme is validated using an analytical solution for the oscillation of a fluid in a para ..."
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Cited by 3 (1 self)
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The wellbalanced, positivitypreserving scheme of Audusse et al, 2004, for the solution of the SaintVenant equations with wetting and drying, is generalised to an adaptive quadtree spatial discretisation. The scheme is validated using an analytical solution for the oscillation of a fluid in a parabolic container, as well as the classic Monai tsunami laboratory benchmark. An efficient database system able to dynamically reconstruct a multiscale bathymetry based on extremely large datasets is also described. This combination of methods is sucessfully applied to the adaptive modelling of the 2004 Indian ocean tsunami. Adaptivity is shown to significantly decrease the exponent of the power law describing computational cost as a function of spatial resolution. The new exponent is directly related to the fractal dimension of the geometrical structures characterising tsunami propagation. The implementation of the method as well as the data and scripts necessary to reproduce the results presented are freely available as part of the opensource Gerris Flow Solver framework. 1
THE RIEMANN PROBLEM FOR THE SHALLOW WATER EQUATIONS WITH DISCONTINUOUS TOPOGRAPHY
, 712
"... Abstract. 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 wav ..."
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Cited by 2 (1 self)
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Abstract. 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. 1.