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182
Compact central WENO schemes for multidimensional conservation laws
 SIAM J. Sci. Comput
, 2000
"... We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greate ..."
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Cited by 59 (12 self)
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We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the thirdorder scheme: one scheme that is based on a genuinely twodimensional Central WENO reconstruction, and another scheme that is based on a simpler dimensionbydimension reconstruction. The simpler dimensionbydimension variant is then extended to a multidimensional fifthorder scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes. Key words. HamiltonJacobi equations, central schemes, high order, WENO, CWENO.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
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Algebraic flux correction I. Scalar conservation laws. Chapter 6 in the first edition of this book
, 2005
"... Abstract This chapter is concerned with the design of highresolution finite element schemes satisfying the discrete maximum principle. The presented algebraic flux correction paradigm is a generalization of the fluxcorrected transport (FCT) methodology. Given the standard Galerkin discretization ..."
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Cited by 45 (23 self)
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Abstract This chapter is concerned with the design of highresolution finite element schemes satisfying the discrete maximum principle. The presented algebraic flux correction paradigm is a generalization of the fluxcorrected transport (FCT) methodology. Given the standard Galerkin discretization of a scalar transport equation, we decompose the antidiffusive part of the discrete operator into numerical fluxes and limit these fluxes in a conservative way. The purpose of this manipulation is to make the antidiffusive term local extremum diminishing. The available limiting techniques include a family of implicit FCT schemes and a new linearitypreserving limiter which provides a unified treatment of stationary and timedependent problems. The use of Anderson acceleration makes it possible to design a simple and efficient quasiNewton solver for the constrained Galerkin scheme. We also present a linearized FCT method for computations with small time steps. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convectiondominated transport problems and anisotropic diffusion equations. 1
Wellbalanced finite volume schemes of arbitrary order of accuracy for shallow water flows
, 2006
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Convergence Of Numerical Schemes For Viscosity Solutions To IntegroDifferential Degenerate Parabolic Problems Arising In Financial Theory
 NUMER. MATH
, 2001
"... We study the numerical approximation of viscosity solutions for integrodifferential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated BlackScholes equation, when the processes which generate the underlying stock returns may contain both ..."
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Cited by 26 (4 self)
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We study the numerical approximation of viscosity solutions for integrodifferential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated BlackScholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.
A lownumerical dissipation patchbased adaptive mesh refinement method for largeeddy simulation of compressible flows
, 2005
"... This paper presents a hybrid finitedi#erence/weighted essentially nonoscillatory (WENO) method for largeeddy simulation of compressible flows with lownumerical dissipation schemes and structured adaptive mesh refinement (SAMR). A conservative fluxbased approach is described, encompassing the ..."
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Cited by 25 (9 self)
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This paper presents a hybrid finitedi#erence/weighted essentially nonoscillatory (WENO) method for largeeddy simulation of compressible flows with lownumerical dissipation schemes and structured adaptive mesh refinement (SAMR). A conservative fluxbased approach is described, encompassing the cases of scheme alternation and internal mesh interfaces resulting from SAMR. An explicit centered scheme is used in turbulent flow regions while a WENO scheme is employed to capture shocks.
Global optimization of explicit strongstabilitypreserving Runge–Kutta methods
 Math. Comp
"... Abstract. Strongstabilitypreserving RungeKutta (SSPRK) methods are a type of time discretization method that are widely used especially for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stab ..."
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Cited by 25 (0 self)
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Abstract. Strongstabilitypreserving RungeKutta (SSPRK) methods are a type of time discretization method that are widely used especially for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE; e.g., positivity or stability with respect to total variation. This is of particular interest when the solution exhibits shocklike or other nonsmooth behaviour. A variety of optimality results have been proven for simple SSPRK methods. However, the scope of these results has been limited to loworder methods due to the detailed nature of the proofs. In this article, global optimization software, BARON [28], is applied to an appropriate mathematical formulation to obtain optimality results for general explicit SSPRK methods up to fifthorder and explicit lowstorage SSPRK methods up to fourthorder. Throughout, our studies allow for the possibility of negative coefficients which correspond to downwindbiased spatial discretizations. Guarantees of optimality are obtained for a variety of third and fourth order schemes. Where optimality is impractical to guarantee (specifically, for fifthorder methods and certain lowstorage methods), extensive numerical optimizations are carried out to derive numerically optimal schemes. As a part of these studies, several new schemes arise which have theoretically improved timestepping restrictions over schemes appearing in the recent literature. 1.
A mathematical model for the SO_2 aggression to calcium carbonate stones: numerical approximation and asymptotic analysis
 IAC Report n. 6
, 2003
"... We introduce a degenerate nonlinear parabolic system which describes the chemical aggression of Calcium Carbonate stones under the attack of SO_2. For this system, we present some finite elements and finite differences schemes to approximate its solutions. Numerical stability is given under suitab ..."
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Cited by 23 (8 self)
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We introduce a degenerate nonlinear parabolic system which describes the chemical aggression of Calcium Carbonate stones under the attack of SO_2. For this system, we present some finite elements and finite differences schemes to approximate its solutions. Numerical stability is given under suitable CFL conditions. Finally, by means of a formal scaling, the qualitative behavior of the solutions for large times is investigated and a numerical verification of this asymptotics is given. Our results are in perfect agreement with the experimental behavior observed in the chemical literature.
Finite volume schemes for dispersive wave propagation and runup
 J. Comput. Phys
, 2011
"... ar ..."
Explicit Diffusive Kinetic Schemes For Nonlinear Degenerate Parabolic Systems
 Parabolic Systems, Math. Comp
, 2000
"... We design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection. These schemes are based on discrete BGK models where both characteristic velocities and the sourceterm depend singularly on the relaxation parameter. General stability conditions are derived, a ..."
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Cited by 20 (3 self)
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We design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection. These schemes are based on discrete BGK models where both characteristic velocities and the sourceterm depend singularly on the relaxation parameter. General stability conditions are derived, and convergence is proved to the entropy solutions for scalar equations.