Exponential Time Differencing for Stiff Systems

By Stephen M. Cox, Paul Matthews. 2002

1,466 citations recorded by the source corpus.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9a47a0c3f84a5aa1ef8016b8d9655951bc830933

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In Computational Mechanics

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Citations in the corpus

Exponential Runge--Kutta methods for parabolic problems (2005, methodology): “Exponential multistep methods were first considered by Nørsett [17] and more recently derived in [3].”
The phase field method for geometric moving interfaces and their numerical approximations (2019, positive): “Example 3: Exponential integrator Another way of utilizing the linear splitting of the J ′(u) = Au+N(u) is to develop exponential integrators [121, 259, 282], The central idea is to use an equivalent formulation of the phase field equation”
Local discontinuous Galerkin methods for nonlinear Schrödinger equations (2005, methodology): “We can also use the exponential time differencing fourth-order Runge–Kutta method which was developed by Cox and Matthews in [16].”
The scalar auxiliary variable (SAV) approach for gradient flows (2018, methodology): “The reference solutions are computed using the fourth order scheme ETDRK4 (see [5]) with sufficiently small time steps.”
Ensemble Kalman methods for inverse problems (2012, methodology): “The method used to approximate the forward model is a modification of a fourth-order Runge-Kutta method, ETD4RK [7], in which the Stokes semi-group is computed exactly by working in the incompressible Fourier basis {ψk(x)}k∈Z2\{0}, and Duhamel’s principle (variation of constants formula) is used to ”
Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations (2009, methodology): “Cox and Matthews derive a set of ETD methods based on Runge-Kutta time-stepping, which they call ETDRK schemes in [43], among general formulas for ETD-schemes to arbitrary order.”
Exponential integrators (2010, positive): “Cox and Matthews (2002) proposed the following exponential variant of the classical Runge–Kutta method: 0 1 2 1 2φ1,2 1 2 0 1 2φ1,3 1 12φ1,3(φ0,3 − 1) 0 φ1,3 φ1 − 3φ2 + 4φ3 2φ2 − 4φ3 2φ2 − 4φ3 4φ3 − φ2 .”
TAYLOR EXPANSIONS OF SOLUTIONS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE (2010, neutral): “, [3-5, 16-19, 27, 30, 31, 36] ).”
Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems (2005, methodology): “Motivated by recent interest in exponential integrators for stiff problems [3] [9] [11] [12] and inspired by the promising numerical experiments reported in these papers, we present error bounds for a class of explicit exponential Runge-Kutta methods for semilinear parabolic problems u (t) + Au(t)…”
An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs (2013, background): “Finally, for comparison we compute the solution using the standard parareal method by solving the full equation (1.1) using big and small step sizes ∆ T and ∆ t , where we use Strang splitting for the coarse solver (we ﬁnd that using ETDRK4 or OIFS as a coarse solver yields similar parallel speedup)”
Finite difference methods for ordinary and partial differential equations - steady-state and time-dependent problems (2007, methodology): “See, for example, [8], [19], [48], [53] for more discussion of these methods.”