On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

Gottlieb, Sigal

doi:10.1007/s10915-004-4635-5

Cited by 162 publications

(141 citation statements)

References 29 publications

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“…Among the various classes of ODE solvers, explicit Runge-Kutta methods have proven to have the best potential for large effective SSP coefficients. Results for SSP multistep methods are disappointing, in the sense that the SSP coefficients are very small [26,9,14,6] (even for implicit multistep methods, c ≤ 2 for methods of higher than first order accuracy). By considering a weaker property wherein particular starting procedures are prescribed, Ruuth and Hundsdorfer [23] have developed methods that are competitive in some cases with optimal Runge-Kutta methods; however, these methods require more memory and in some cases were observed to violate the SSP property.…”

Section: |U(t + τ )|| ≤ ||U(t)|| ∀τ ≥ 0 (12)mentioning

confidence: 99%

Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations

Ketcheson¹

2008

SIAM J. Sci. Comput.

163

222

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Abstract. Strong stability-preserving (SSP) Runge-Kutta methods were developed for time integration of semidiscretizations of partial differential equations. SSP methods preserve stability properties satisfied by forward Euler time integration, under a modified time-step restriction. We consider the problem of finding explicit Runge-Kutta methods with optimal SSP time-step restrictions, first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations. By using alternate formulations of the associated optimization problems and introducing a new, more general class of low-storage implementations of Runge-Kutta methods, new optimal low-storage methods and new low-storage implementations of known optimal methods are found. The results include families of low-storage second and third order methods that achieve the maximum theoretically achievable effective SSP coefficient (independent of stage number), as well as low-storage fourth order methods that are more efficient than current full-storage methods. The theoretical properties of these methods are confirmed by numerical experiment.

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Section: |U(t + τ )|| ≤ ||U(t)|| ∀τ ≥ 0 (12)mentioning

confidence: 99%

Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations

Ketcheson¹

2008

SIAM J. Sci. Comput.

163

222

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“…SSP methods preserve the strong stability properties -in any norm, seminorm or convex functional -of the spatial discretization coupled with first order Euler time stepping, when the timestep is suitably restricted. Explicit strong stability preserving (SSP) Runge-Kutta methods ([17] [6]) have been successfully used with a wide range of spatial discretizations, including spectral, discontinuous Galerkin, and weighted essentially non-oscillatory (WENO) methods. These high order methods preserve any nonlinear stability properties satisfied by the spatial discretization coupled with the forward Euler time-stepping.…”

Section: Summary Of Aims and Resultsmentioning

confidence: 99%

“…case 'SSP104' s=10; r=6; alphaO=diag(ones(l,s-l),-1); alpha0(6,5)=2/5; alpha0(6,l)=3/5; betaO =l/6*diag(ones(l,s-l),-1); beta0 (6,5) (6))/18 (-l+sqrt (6))/18 1/9 (88+7*sqrt (6))/360 (88-43*sqrt(6))/360 35 1/9 (88+43*sqrt (6))/360 (88-7*sqrt(6))/360]; b= [l/9 (16+sqrt(6))/36 (16-sqrt(6))/36]'; c=[0 (6-sqrt(6))/10 (6+sqrt (6) (6))/360 (296-169*sqrt (6))/1800 (-2+3*sqrt (6))/225 (296+169*sqrt (6))/1800 (88+7*sqrt (6))/360 (-2-3*sqrt (6))/225 (16-sqrt (6))/36 (16+sqrt (6))/36 1/9]; b= [(16-sqrt(6))/36 (16+sqrt (6) acc=l. …”

Section: Appendixmentioning

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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“…(2.46)-(2.48), a number of StrongStability-Preserving (SSP) explicit time discretization methods are available [Got05]. The Total Variation Diminishing (TVD) Runge-Kutta methods of Shu and Osher [SO89] (a subclass of SSP) are particularly suited for this purpose.…”

Section: Explicit Runge-kutta Schemementioning

confidence: 99%

Solution Algorithms for Effective-Field Models of Multi-Fluid Flows

Christon¹

2012

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On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

Cited by 162 publications

References 29 publications

Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations

Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations

Implicit High Order Strong Stability Preserving Runge-Kutta Time Discretizations

Solution Algorithms for Effective-Field Models of Multi-Fluid Flows

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