Results 11  20
of
156
Highorder multiimplicit spectral deferred correction methods for problems of reactive flow,
 J. Comput. Phys.
, 2003
"... Abstract Models for reacting flow are typically based on advectiondiffusionreaction (ADR) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three subprocesses can be widely different, leading to disparate tim ..."
Abstract

Cited by 26 (9 self)
 Add to MetaCart
(Show Context)
Abstract Models for reacting flow are typically based on advectiondiffusionreaction (ADR) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three subprocesses can be widely different, leading to disparate timestep requirements for robust and accurate timeintegration. In particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this timescale disparity. The proposed methods are highorder multiimplicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of ADR equations. Spectral deferred correction methods compute a highorder approximation to the solution of a differential equation by using a simple, loworder numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several subprocesses implicitly but independently, while avoiding the splitting errors present in traditional operatorsplitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semiimplicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the subprocess time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific timescales ordering, the generalization to any ordering combination is straightforward.
Exponential RungeKutta methods for stiff kinetic equations
 SIAM J. Num. Anal
, 2011
"... ar ..."
(Show Context)
Multisymplectic box schemes and the Korteweg–de Vries equation
, 2003
"... We develop and compare some geometric integrators for the Kortewegde Vries equation, especially with regard to their robustness for large steps in space and time, ∆x and ∆t, and over long times. A standard, semiexplicit, symplectic finite difference scheme is found to be fast and robust. However, ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
We develop and compare some geometric integrators for the Kortewegde Vries equation, especially with regard to their robustness for large steps in space and time, ∆x and ∆t, and over long times. A standard, semiexplicit, symplectic finite difference scheme is found to be fast and robust. However, in some parameter regimes such schemes are susceptible to developing small wiggles. At the same instances the fully implicit and multisymplectic Preissmann scheme, written as a 12point box scheme, stays smooth. This is accounted for by the ability of the box scheme to preserve the shape of the dispersion relation of any hyperbolic system for all ∆x and ∆t. We also develop a simplified 8point version of this box scheme which maintains its advantageous features.
Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations
 BIT
"... This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn, tn+1] into substeps, and use function values comput ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
(Show Context)
This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn, tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between GaussLegendre, GaussLobatto, GaussRadau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the lefthand endpoint tn. Quadrature rules that do not depend on the lefthand endpoint (which are referred to as righthand quadrature rules) are shown to lead to L(α)stable implicit methods with α ≈ π/2. The semiimplicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders
ImplicitExplicit Numerical Schemes for JumpDiffusion Processes
 Calcolo
, 2004
"... We study the numerical approximation of viscosity solutions for Parabolic IntegroDifferential Equations (PIDE). Similar models arise in option pricing, to generalize the BlackScholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jum ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
(Show Context)
We study the numerical approximation of viscosity solutions for Parabolic IntegroDifferential Equations (PIDE). Similar models arise in option pricing, to generalize the BlackScholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the nonlocal nature of the integral term, unconditionally stable implicit difference scheme are not practically feasible. Here we propose to use ImplicitExplicit (IMEX) RungeKutta methods for the time integration to solve the integral term explicitly, giving higher order accuracy schemes under weak stability timestep restrictions. Numerical tests are presented to show the computational efficiency of the approximation.
IMEX RungeKutta schemes and hyperbolic systems of conservation laws with stiff diffusive relaxation
 ICNAAM, AIP Conference Proceedings 1168
, 2009
"... ar ..."
(Show Context)
Central schemes on overlapping cells
, 2005
"... Nessyahu and Tadmor's central scheme [J. Comput. Phys. 87 (1990)] has the benefit of not using Riemann solvers for solving hyperbolic conservation laws. But the staggered averaging causes large dissipation when the time step size is small compared to the mesh size. The recent work of Kurganov ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
Nessyahu and Tadmor's central scheme [J. Comput. Phys. 87 (1990)] has the benefit of not using Riemann solvers for solving hyperbolic conservation laws. But the staggered averaging causes large dissipation when the time step size is small compared to the mesh size. The recent work of Kurganov and Tadmor [J. Comput. Phys. 160 (2000)] overcomes this problem by using a variable control volume and results in semidiscrete and fully discrete nonstaggered schemes. Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the O(1/Dt) dependence of the dissipation. The semidiscrete form of the central scheme can also be obtained to which the TVD Runge–Kutta time discretization methods of Shu and Osher [J. Comput. Phys. 77 (1988)] or other stable and sufficiently accurate ODE solvers can be applied. This technique is essentially independent of the reconstruction and the shape of the mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions. Generally speaking, more compact reconstruction can be achieved. In the following, schemes of up to fifth order in 1D and third order in 2D have been developed. We demonstrate through numerical examples that by combining two classes of the overlapping cells in the reconstruction we can achieve higher resolution.
The implicit Closest Point Method for the numerical solution of partial differential equations on surfaces
 SIAM J. Sci. Comput
"... Abstract. Many applications in the natural and applied sciences require the solutions of partial differential equations (PDEs) on surfaces or more general manifolds. The Closest Point Method is a simple and accurate embedding method for numerically approximating PDEs on rather general smooth surface ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
(Show Context)
Abstract. Many applications in the natural and applied sciences require the solutions of partial differential equations (PDEs) on surfaces or more general manifolds. The Closest Point Method is a simple and accurate embedding method for numerically approximating PDEs on rather general smooth surfaces. However, the original formulation is designed to use explicit time stepping. This may lead to a strict timestep restriction for some important PDEs such as those involving the LaplaceBeltrami operator or higherorder derivative operators. To achieve improved stability and efficiency, we introduce a new implicit Closest Point Method for surface PDEs. The method allows for large, stable time steps while retaining the principal benefits of the original method. In particular, it maintains the order of accuracy of the discretization of the underlying embedding PDE, it works on sharply defined bands without degrading the accuracy of the method, and it applies to general smooth surfaces. It also is very simple and may be applied to a rather general class of surface PDEs. Convergence studies for the insurface heat equation and a fourthorder biharmonic problem are given to illustrate the accuracy of the method. We demonstrate the flexibility and generality of the method by treating flows involving diffusion, reactiondiffusion and fourthorder spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension. Key words. Closest Point Method, surface computation, implicit surfaces, partial differential equations, implicit time stepping, Laplace–Beltrami operator, biharmonic operator, surface diffusion AMS subject classifications. 65M06, 58J35, 65M20 1. Introduction. Partial
IMPLICITEXPLICIT VARIATIONAL INTEGRATION OF HIGHLY OSCILLATORY PROBLEMS
, 808
"... ABSTRACT. In this paper, we derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level of the differential equations or the Hamiltonian, w ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
ABSTRACT. In this paper, we derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level of the differential equations or the Hamiltonian, we split the two potentials with respect to the Lagrangian action integral. By using a different quadrature rule to approximate the contribution of each potential to the action, we arrive at a geometric integrator that is implicit in the fast force and explicit in the slow force. This can allow for significantly longer time steps to be taken (compared to standard explicit methods, such as Störmer/Verlet) at the cost of only a linear solve rather than a full nonlinear solve. We also analyze the stability of this method, in particular proving that it eliminates the linear resonance instabilities that can arise with explicit multipletimestepping methods. Next, we perform some numerical experiments, studying the behavior of this integrator for two test problems: a system of coupled linear oscillators, for which we compare against the resonance behavior of the rRESPA method; and slow energy exchange in the Fermi–Pasta–Ulam problem, which couples fast linear oscillators with slow nonlinear oscillators. Finally, we prove that this integrator accurately preserves the slow energy exchange between the fast oscillatory components, which explains the numerical behavior observed for the Fermi–Pasta–Ulam problem. 1.