Positivity of Runge-Kutta and diagonally split Runge-Kutta methods

Horváth, Zoltán

doi:10.1016/s0168-9274(98)00050-6

Cited by 68 publications

(42 citation statements)

References 11 publications

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“…In a numerical solution, the function κ yields a stepsize condition, under which the considered discretization scheme gives a nonnegative approximation x ∆ ≥ 0 of the exact solution x ≥ 0, cp. e.g., [13,61], [76,77,78], [80, pp. 122] and the references therein.…”

Section: Ordinary Differential Equationsmentioning

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%

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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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Section: Ordinary Differential Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Baum

2015

J Dyn Diff Equat

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“…We will use the following simple but very convenient criterion on F that tells us whether the system is positive, see [7,15,16]: Suppose that F ( ) is continuous and satisfies the Lipschitz condition with respect to . Then system (6) is positive if and only if for any vector ∈ R and all = 1 and ≥ 0,…”

Section: A Priori Estimates Of the Solutionmentioning

confidence: 99%

Numerical analysis of nonlinear model of excited carrier decay

2013

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This paper presents a mathematical model for photo-excited carrier decay in a semiconductor. Due to the carrier trapping states and recombination centers in the bandgap, the carrier decay process is defined by the system of nonlinear differential equations. The system of nonlinear ordinary differential equations is approximated by linearized backward Euler scheme. Some a priori estimates of the discrete solution are obtained and the convergence of the linearized backward Euler method is proved. The identifiability analysis of the parameters of deep centers is performed and the fitting of the model to experimental data is done by using the genetic optimization algorithm. Results of numerical experiments are presented. MSC:65L12, 65L20, 65Z05

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“…"A numerical study of diagonally split Runge-Kutta methods for PDEs with discontinuities" by C. 2,8,9]) time discretization methods are a class of implicit time-stepping schemes which offer both (formal) high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge-Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs.…”

Section: Publicationsmentioning

confidence: 99%

Implicit High Order Strong Stability Preserving Runge-Kutta Time Discretizations

Gottlieb

2009

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this burden to Department of Defense, Washington Headquarters Services, Directorate for Information , 121 4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply wnn a collection ot intormanon it it does not display a currently valid OMB control number PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. REPORT DATE (DD-MM-YYYY) SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION / AVAILABILITY STATEMENTDistribution A: Approved for Public Release SUPPLEMENTARY NOTES ABSTRACTThis research involved the investigation, development, and testing of diagonally split RungeKutta (DSRK) methods and implicit Runge-Kutta methods with reference to their strong stability preserving (SSP) properties for large time-steps. The research found that DSRK methods which are unconditionally SSP reduce to first order for the stepsizes of interest, and the PI introduced an analysis which explains this phenomenon and shows that it is unavoidable. The PI and her students developed a methodology for finding optimal implicit SSP Runge--Kutta methods up to order six (which is the maximal possible order for these methods) and eleven stages, and found that the effective SSP coefficient can be no more than two, making these methods not competitive with explicit methods for most applications, but useful in a carefully chosen subset of problems. We investigated diagonally split Runge-Kutta (DSRK) methods to identify and test unconditionally strong stability preserving (SSP) methods, and implicit SSP timestopping methods to find methods with a large SSP coefficient. We found that DSRK methods which are unconditionally SSP reduce to first order for the stepsizes of interest, and introduced an analysis which explains this phenomenon and shows that it is unavoidable. We found optima; implicit SSP Runge-Kutta methods up to order six (which is the maximal possible order for these methods) and eleven stages, and found that the effective SSP coefficient can be no more than two, making these methods not competitive with explicit methods for most applications, but useful in a carefully chosen subset of problems. We now have a complete analysis of implicit SSP RungeKutta methods and demonstrations of the need for the SSP property in solutions of hyperbolic PDEs with shocks. Summary of Aims and ResultsStrong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties -in any norm, seminorm or convex functional -of the spatial discretization c...

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Positivity of Runge-Kutta and diagonally split Runge-Kutta methods

Cited by 68 publications

References 11 publications

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

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

Numerical analysis of nonlinear model of excited carrier decay

Implicit High Order Strong Stability Preserving Runge-Kutta Time Discretizations

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