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156
Fourthorder time stepping for stiff PDEs
 SIAM J. SCI. COMPUT
, 2005
"... A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made ..."
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Cited by 94 (3 self)
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A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential timedifferencing (ETD) scheme against the competing methods of implicitexplicit differencing, integrating factors, timesplitting, and Fornberg and Driscoll’s “sliders ” for the KdV, Kuramoto–Sivashinsky, Burgers, and Allen–Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.
ImplicitExplicit RungeKutta schemes and applications to hyperbolic systems with relaxation
 Journal of Scientific Computing
, 2000
"... We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The sch ..."
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Cited by 83 (14 self)
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We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by finite difference discretization with Weighted Essentially Non Oscillatory (WENO) reconstruction. After a brief description of the mathematical properties of the schemes, several applications will be presented. Keywords: RungeKutta methods, hyperbolic systems with relaxation, stiff systems, high order shock capturing schemes. AMS Subject Classification: 65C20, 82D25 1
Semiimplicit spectral deferred correction methods for ordinary differential equations.
 Commun. Math. Sci.
, 2002
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Efficient Simulation of Inextensible Cloth
"... Many textiles do not noticeably stretch under their own weight. Unfortunately, for better performance many cloth solvers disregard this fact. We propose a method to obtain very low strain along the warp and weft direction using Constrained Lagrangian Mechanics and a novel fast projection method. The ..."
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Cited by 52 (3 self)
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Many textiles do not noticeably stretch under their own weight. Unfortunately, for better performance many cloth solvers disregard this fact. We propose a method to obtain very low strain along the warp and weft direction using Constrained Lagrangian Mechanics and a novel fast projection method. The resulting algorithm acts as a velocity filter that easily integrates into existing simulation code. CR Categories: I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism—Animation I.6.8 [Simulation and Modeling]:
Recurrent motions within plane Couette turbulence
 Journal of Fluid Mechanics
"... We describe accurate computations of threedimensional periodic and relative periodic motions within plane Couette turbulence at Re = 400. To ensure that the computed solutions are true solutions of the NavierStokes equations, careful attention is paid to time discretization errors and to spatial r ..."
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Cited by 50 (5 self)
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We describe accurate computations of threedimensional periodic and relative periodic motions within plane Couette turbulence at Re = 400. To ensure that the computed solutions are true solutions of the NavierStokes equations, careful attention is paid to time discretization errors and to spatial resolution. All the computed solutions are linearly unstable. While direct numerical simulation helps us understand the statistics of turbulent fluid flows, elucidation of the geometry of turbulent flows in phase space requires the computation of steady states, traveling waves, periodic motions, and close recurrences. The computed solutions are used as a basis to discuss the manner in which the geometry of turbulent dynamics in phase space can be understood. The method used for computing these solutions is described in detail.
A fast spectral algorithm for nonlinear wave equations with linear dispersion
 J. Comput. Phys
, 1999
"... Spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains, including wellknown cases such as the Korteweg–de Vries and nonlinear Schrödinger equations. For the best computational efficiency, one needs also to use highorder methods in ti ..."
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Cited by 47 (5 self)
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Spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains, including wellknown cases such as the Korteweg–de Vries and nonlinear Schrödinger equations. For the best computational efficiency, one needs also to use highorder methods in time while somehow bypassing the usual severe stability restrictions. We use linearly implicit multistep methods, with the innovation of choosing different methods for different ranges in Fourier space—high accuracy at low wavenumbers and Astability at high wavenumbers. This new approach compares favorably to alternatives such as splitstep and integrating factor (or linearly exact) methods. c ○ 1999 Academic Press Key Words: spectral methods; nonlinear waves; KdV; NLS; linearly implicit. 1.
Analysis of Stiffness in the Immersed Boundary Method and Implications for Timestepping Schemes
 J. Comput. Phys
, 1998
"... The immersed boundary method is known to exhibit a high degree of numerical stiffness, which is associated with the interaction of immersed elastic fibres with the surrounding fluid. We perform a linear analysis of the underlying equations of motion for immersed fibres, and identify a discrete set ..."
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Cited by 39 (1 self)
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The immersed boundary method is known to exhibit a high degree of numerical stiffness, which is associated with the interaction of immersed elastic fibres with the surrounding fluid. We perform a linear analysis of the underlying equations of motion for immersed fibres, and identify a discrete set of fibre modes which are associated solely with the presence of the fibre. These results are a generalisation of those in a previous paper (SIAM J. Appl. Math., 55(6):15771591, 1995) by including the effect of spreading the singular fibre force over a finite "smoothing radius," which corresponds to the approximate delta function used in the immersed boundary method. We investigate the stability of the fibre modes, their stiffness and dependence on the problem parameters, and the effect that smoothing has on the solution. The analytical results are then extended to include the effects of time discretisation, and conclusions are drawn about the time step restrictions on various expli...
Numerical time integration for air pollution models
 SURVEYS ON MATHEMATICS FOR INDUSTRY
, 1998
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Fiberdyne Systems
 Proceedings of FOC '78, Information Gatekeepers
"... SemiLagrangian multistep exponential integrators for index 2 differential algebraic system by ..."
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Cited by 29 (0 self)
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SemiLagrangian multistep exponential integrators for index 2 differential algebraic system by
Central Schemes For Balance Laws Of Relaxation Type
 SIAM J. NUMER. ANAL
, 2000
"... Several models in mathematical physics are described by quasilinear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodyna ..."
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Cited by 28 (7 self)
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Several models in mathematical physics are described by quasilinear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodynamics are presented. The numerical methods are a generalization of the NessyahuTadmor scheme to the nonhomogeneous case byincluding the cell averages of the production terms in the discrete balance equations. A second order scheme uniformlyaccurate in the relaxation parameter is derived and its properties analyzed. Numerical tests confirm the accuracy and robustness of the scheme.