Computation of optimal monotonicity preserving general linear methods

Ketcheson, David I.

doi:10.1090/s0025-5718-09-02209-1

Cited by 30 publications

(40 citation statements)

References 33 publications

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“…This requirement leads to severe restrictions on the allowable order of SSP methods, and the allowable time step ∆t ≤ C∆t FE . These methods have been extensively studied, e.g., in [20,21,22,27,28,29,33,34,36,39,40,41,42,45,65]. In this section, we review some popular and efficient explicit SSP Runge-Kutta methods, and present the SSP coefficients of optimized methods of up to ten stages and fourth order.…”

Section: A Review Of Explicit Ssp Runge-kutta Methodsmentioning

confidence: 99%

Strong Stability Preserving Integrating Factor Runge--Kutta Methods

Isherwood¹,

Grant²

2018

SIAM J. Numer. Anal.

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Strong stability preserving (SSP) Runge-Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-explicit methods have very restrictive time-steps and are therefore not efficient. For this reason, SSP integrating factor methods may offer an attractive alternative to traditional time-stepping methods for problems with a linear component that is stiff and a nonlinear component that is not. However, the strong stability properties of integrating factor Runge-Kutta methods have not been established. In this work we show that it is possible to define explicit integrating factor Runge-Kutta methods that preserve the desired strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping, or even given weaker conditions. We define sufficient conditions for an explicit integrating factor Runge-Kutta method to be SSP, namely that they are based on explicit SSP Runge-Kutta methods with non-decreasing abscissas. We find such methods of up to fourth order and up to ten stages, analyze their SSP coefficients, and prove their optimality in a few cases. We test these methods to demonstrate their convergence and to show that the SSP time-step predicted by the theory is generally sharp, and that the non-decreasing abscissa condition is needed in our test cases. Finally, we show that on typical total variation diminishing linear and nonlinear test-cases our new explicit SSP integrating factor Runge-Kutta methods out-perform the corresponding explicit SSP Runge-Kutta methods, implicit-explicit SSP Runge-Kutta methods, and some well-known exponential time differencing methods.

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Section: A Review Of Explicit Ssp Runge-kutta Methodsmentioning

confidence: 99%

Strong Stability Preserving Integrating Factor Runge--Kutta Methods

Isherwood¹,

Grant²

2018

SIAM J. Numer. Anal.

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“…By using bisection in r, the optimization problem (3.10) can be viewed as a sequence of linear feasible problems, as suggested in [10]. We solved the above problem using linprog in Matlab and found optimal IMEX SSP methods for k ∈ {1, .…”

Section: Monotone Imex Linear Multistep Methodsmentioning

confidence: 99%

“…Optimal explicit linear multistep schemes of order up to six, coupled with efficient upwind and downwind WENO discretizations, were studied in [4]. Coefficients of optimal upwind-and downwind-biased methods together with a reformulation of the nonlinear optimization problem involved as a series of linear programming feasibility problems can be found in [10]. Bounds on the maximum SSP step size for downwind-biased methods have been analyzed in [11].…”

Section: Introductionmentioning

confidence: 99%

Strong-stability-preserving additive linear multistep methods

Hadjimichael

Ketcheson

2018

Math. Comp.

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The analysis of strong-stability-preserving (SSP) linear multistep methods is extended to semi-discretized problems for which different terms on the right-hand side satisfy different forward Euler (or circle) conditions. Optimal additive and perturbed monotonicitypreserving linear multistep methods are studied in the context of such problems. Optimal perturbed methods attain larger monotonicity-preserving step sizes when the different forward Euler conditions are taken into account. On the other hand, we show that optimal SSP additive methods achieve a monotonicity-preserving step-size restriction no better than that of the corresponding non-additive SSP linear multistep methods.

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“…[80, p. 123]. The absolute monotonicity radius R (A,β,γ) and its computation are analyzed, e.g., in [59,61,68,87,129] and [100,101,84] and the references therein, where R (A,β,γ) is discussed in the context of contractivity preserving one-and multistep methods.…”

Section: Characterization Of Other Flow Propertiesmentioning

confidence: 99%

“…[20,19,18,45,47,80,83,113,114,126] and the references therein. Numerical methods that preserve the property of positivity within the discretization have been discussed in [13,76,77,78,80] and in the context of stability, contractivity or monotonicity preserving methods in [59,61,68,84,87,100,101,129]. Due to balance equations, conservation laws or limitation of resources, these processes often are subject to additional constraints leading to the notion of positive DAEs.…”

Section: Introductionmentioning

confidence: 99%

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

Baum

2015

J Dyn Diff Equat

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Differential-algebraic equations (DAEs) are coupled systems of differential and algebraic equations arising in the modeling of dynamic processes that are restricted by auxiliary, algebraic constraints. Examples are, e.g., multibody systems, where the movement of the system is forced to a predefined trajectory by connected joints, electrical circuits or networks, where connections and loops lead to algebraic relations of the flow and effort variables, or biological systems, reaction networks or advection-diffusion equations, where the evolution of the process is restricted by balance equations and conservation laws. Modeling dynamic processes in economic or social sciences, in biological or chemical engineering, the analyzed values typically represent real-valued quantities like the amount of goods or individuals or the density of a chemical or biological species that cannot take negative values. In combination with the algebraic constraints, this property leads to the notion of positive DAEs, i.e., systems whose solutions remain componentwise nonnegative whenever the initial value is nonnegative.In this work, we generalize the concept of the flow from ordinary differential equations (ODEs) to DAEs and use this framework to study system properties like invariant sets and positivity. To decouple the differential and algebraic components in a DAE, we develop an approach that allows to parameterize embedded submanifolds using projections and that extends the idea of projections to embedded submanifold. Combined with the theory of the strangeness-index, we apply this projection approach to remodel a given DAE as a set of explicit differential and algebraic equations. Solving the decoupled system and showing existence and uniqueness of the solution representation, we generalize the notion of the flow to DAEs. For linear systems, we generalize Duhamel's formula. Based on the flow, we extend the results of invariant sets from ODEs to DAEs and we characterize linear and nonlinear DAEs with regard to positivity. ZUSAMMENFASSUNG

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Computation of optimal monotonicity preserving general linear methods

Cited by 30 publications

References 33 publications

Strong Stability Preserving Integrating Factor Runge--Kutta Methods

Strong Stability Preserving Integrating Factor Runge--Kutta Methods

Strong-stability-preserving additive linear multistep methods

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

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