David Ketcheson

8577669100

Publications - 4

Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method

Tihamér A. Kocsis Inmaculada Higueras David Ketcheson

Publication Name: Journal of Scientific Computing

Publication Date: 2018-09-01

Volume: 76

Issue: 3

Page Range: 1337-1369

Description:

Perturbed Runge–Kutta methods (also referred to as downwind Runge–Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge–Kutta counterparts. In this paper we study the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.

Open Access: Yes

DOI: 10.1007/s10915-018-0664-3

EXPLICIT strong stability preserving multistep runge-kutta methods

Christopher Bresten Daniel Higgs Zachary Grant Adrián Németh Sigal Gottlieb David Ketcheson

Publication Name: Mathematics of Computation

Publication Date: 2017-01-01

Volume: 86

Issue: 304

Page Range: 747-769

Description:

High-order spatial discretizations of hyperbolic PDEs are often designed to have strong stability properties, such as monotonicity. We study explicit multistep Runge-Kutta strong stability preserving (SSP) time integration methods for use with such discretizations. We prove an upper bound on the SSP coefficient of explicit multistep Runge-Kutta methods of order two and above. Numerical optimization is used to find optimized explicit methods of up to five steps, eight stages, and tenth order. These methods are tested on the linear advection and nonlinear Buckley-Leverett equations, and the results for the observed total variation diminishing and/or positivity preserving time-step are presented.

Open Access: Yes

DOI: 10.1090/mcom/3115

Strong stability preserving explicit linear multistep methods with variable step size

Yiannis Hadjimichael Adrián Németh David Ketcheson Lajos Lóczi

Publication Name: SIAM Journal on Numerical Analysis

Publication Date: 2016-01-01

Volume: 54

Issue: 5

Page Range: 2799-2832

Description:

Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods (of order two and three) with variable step size, and prove their optimality, stability, and convergence. The choice of step size for multistep SSP methods is an interesting problem because the allowable step size depends on the SSP coefficient, which in turn depends on the chosen step sizes. The description of the methods includes an optimal step-size strategy. We prove sharp upper bounds on the allowable step size for explicit SSP linear multistep methods and show the existence of methods with arbitrarily high order of accuracy. The effectiveness of the methods is demonstrated through numerical examples.

Open Access: Yes

DOI: 10.1137/15M101717X

On the absolute stability regions corresponding to partial sums of the exponential function

Tihamér A. Kocsis David Ketcheson Lajos Lóczi

Publication Name: IMA Journal of Numerical Analysis

Publication Date: 2015-07-01

Volume: 35

Issue: 3

Page Range: 1426-1455

Description:

Certain numerical methods for initial value problems have as stability function the nth partial sum of the exponential function. We study the stability region, that is, the set in the complex plane over which the nth partial sum has at most unit modulus. It is known that the asymptotic shape of the part of the stability region in the left half-plane is a semidisc. We quantify this by providing discs that enclose or are enclosed by the stability region or its left half-plane part. The radius of the smallest disc centred at the origin that contains the stability region (or its portion in the left half-plane) is determined for 1 ≤ n ≤ 20. Bounds on such radii are proved for n ≥ 2; these bounds are shown to be optimal in the limit n → +∞. We prove that the stability region and its complement, restricted to the imaginary axis, consist of alternating intervals of length tending to π, as n → ∞. Finally, we prove that a semidisc in the left half-plane centred at the origin and with vertical boundary lying on the imaginary axis is included in the stability region if and only if n ≡ 0 mod 4 or n ≡ 3 mod 4. The maximal radii of such semidiscs are exactly determined for 1 ≤ n ≤ 20.

Open Access: Yes

DOI: 10.1093/imanum/dru039