Monotonicity Conditions for Multirate and Partitioned Explicit Runge-Kutta Schemes

Hundsdorfer, Willem; Mozartova, Anna; Savcenco, Valeriu

doi:10.1007/978-3-642-33221-0_11

Cited by 9 publications

(16 citation statements)

References 19 publications

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“…It was shown in [6] that the conditions for internal consistency (2.4) and conservation of linear invariants (2.6) are incompatible for such multirate schemes.…”

Section: Order Conditionsmentioning

confidence: 99%

“…We consider some explicit multirate schemes that were discussed in [6]; additional examples can be found e.g. in [3,4,15,16,19].…”

Section: Examplesmentioning

confidence: 99%

“…For conservation laws this means that accuracy is studied away from shocks. (For results of multirate schemes near shocks, where monotonicity properties and maximum principles are important, we refer to [6].) The convergence results are illustrated using some existing first-and second-order multirate schemes, applied to semi-discretizations of hyperbolic conservation laws.…”

Section: Introductionmentioning

confidence: 99%

“…This scheme has been described in [6]; it was obtained by adaptation of an implicit (Rosenbrock) scheme from [14]. Although it looks already a bit complicated, the idea is simple: first a coarse Δt step is taken with the explicit trapezoidal rule on the whole index set I , and then two refined 1 2 Δt steps are taken on I 2 , using information from the coarse step by quadratic (Hermite) interpolation at the time level…”

mentioning

confidence: 99%

“…To write the difference of (4.4) and (2.3) in a compact form, let also ρ n = [ρ n,i ] and 6) where the amplification matrix R(Z ) ∈ R m×m and r(Z ) T ∈ R m×ms are defined by…”

mentioning

confidence: 99%

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Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

2014

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An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate.

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“…It was shown in [6] that the conditions for internal consistency (2.4) and conservation of linear invariants (2.6) are incompatible for such multirate schemes.…”

Section: Order Conditionsmentioning

confidence: 99%

“…We consider some explicit multirate schemes that were discussed in [6]; additional examples can be found e.g. in [3,4,15,16,19].…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…To write the difference of (4.4) and (2.3) in a compact form, let also ρ n = [ρ n,i ] and 6) where the amplification matrix R(Z ) ∈ R m×m and r(Z ) T ∈ R m×ms are defined by…”

mentioning

confidence: 99%

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Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

2014

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Multirate generalized additive Runge Kutta methods

Günther

Sandu

2015

Numer. Math.

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This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods (Sandu and Günther, SIAM J Numer Anal, 53(1): 2015). Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many wellknown multirate schemes as special cases. The order conditions theory follows directly from the GARK accuracy theory. Nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold.Mathematics Subject Classification 65L05 · 65L06 · 65L07 · 65L020

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Many-Stage Optimal Stabilized Runge–Kutta Methods for Hyperbolic Partial Differential Equations

Doehring,

Gassner,

Torrilhon

2024

J Sci Comput

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A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge–Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found for non-uniformly refined meshes and spatially varying wave speeds. To demonstrate the applicability of the stability polynomials we construct 2N-storage many-stage Runge–Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. These methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.

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Monotonicity Conditions for Multirate and Partitioned Explicit Runge-Kutta Schemes

Cited by 9 publications

References 19 publications

Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

Multirate generalized additive Runge Kutta methods

Many-Stage Optimal Stabilized Runge–Kutta Methods for Hyperbolic Partial Differential Equations

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