Steady State and Sign Preserving Semi-Implicit Runge--Kutta Methods for ODEs with Stiff Damping Term

Chertock, Alina; Cui, Shumo; Kurganov, Alexander; Wu, Tong

doi:10.1137/151005798

Cited by 41 publications

(48 citation statements)

References 20 publications

(33 reference statements)

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“…We therefore have proved that the steady state is preserved. □ Remark Notice that the 2‐D version of Theorem can be proved in a similar manner. Remark In , we have proved that the time step restriction for the SI‐RK3 method is determined by the nonstiff (explicitly treated) term only and no extra time restrictions due to the stiffness of the friction term is required. This implies that for the ODE system , arising from the central‐upwind semi‐discretization of the 1‐D shallow water system, the size of the time step is to be calculated based on the CFL condition, namely, we need to select

normalΔt \leq {(normalΔt)}_{*} : = \frac{normalΔx}{2 a}, a = \max_{j} a_{j + \frac{1}{2}}^{+}, - a_{j + \frac{1}{2}}^{-},

where

a_{j + \frac{1}{2}}^{\pm}

are the local propagation speeds defined in . Remark We would like to emphasize that , Theorem 2.1] directly applies to the first stage of the SI‐RK3 method (3.25), and hence, the time step restriction guarantees the positivity of

{\overset{h}{false}}_{j}^{I} : = {\overset{w}{false}}_{j}^{I} - B_{j}

for all j provided

{\overset{h}{false}}_{j} (t) \geq 0

for all j .…”

Section: Methodsmentioning

confidence: 67%

“…In these methods, only a portion of the stiff term is implicitly treated, and therefore, the evolution equation is very easy to solve and implement compared to fully implicit or implicit‐explicit methods. The important feature of the ODE solvers we introduced in resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only. The new semi‐implicit methods are based on the modification of explicit SSP‐RK methods and are proven to have a formal second order of accuracy, A ( α )‐stability and stiff decay.…”

Section: Methodsmentioning

confidence: 99%

“…We now implement the SI‐RK3 method to the system (the SI‐RK3 method is a second‐order semi‐implicit Runge‐Kutta method based on the third‐order SSP‐RK method; for details, see , Section 3]). The resulting fully discrete scheme can be written as

\begin{matrix} w_{j}^{I} & = {\overset{true¯}{w}}_{j} (t) + Δt scriptF (1) {[\overset{true¯}{U} (t)]}_{j}, \\ q_{j}^{I} & = \frac{{\overset{true¯}{q}}_{j} (t) + Δt scriptF (2) {[\overset{true¯}{U} (t)]}_{j}}{1 - Δt G (\overset{true¯}{U} {(t)}_{j})}, \end{matrix}

B_{j, k} : = B (x_{j}, y_{k}) = \frac{1}{x y} B (x, y) d x d y = \frac{1}{4} (B_{j + \frac{1}{2}, k} + B_{j - \frac{1}{2}, k} + B_{j, k + \frac{1}{2}} + B_{j, k - \frac{1}{2}})

\begin{matrix} \frac{d}{dt} {\overset{w}{false}}_{j} & = F (1) {[\overset{bold-italicU}{false}]}_{j}, \\ \frac{d}{dt} {\overset{q}{false}}_{j} & = \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

“…We now implement the SI-RK3 method to the system (3.24) (the SI-RK3 method is a secondorder semi-implicit Runge-Kutta method based on the third-order SSP-RK method; for details, see [31,Section 3]). The resulting fully discrete scheme can be written as…”

Section: Time Integration Methodsmentioning

confidence: 99%

“…In [31], we have proved that the time step restriction for the SI-RK3 method is determined by the nonstiff (explicitly treated) term only and no extra time restrictions due to the stiffness of the friction term is required. This implies that for the ODE system (3.24), arising from the central-upwind semidiscretization of the 1-D shallow water system, the size of the time step is to be calculated based on the CFL condition, namely, we need to select…”

Section: Remark 35mentioning

confidence: 99%

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Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

Chertock

Cui

Kurganov

et al. 2015

Numerical Methods in Fluids

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Shallow water models are widely used to describe and study free-surface water flow. Even though, in many practical applications the bottom friction does not have much influence on the solutions, the friction terms will play a significant role when the depth of the water is very small. In this paper, we study the Saint-Venant system of shallow water equations with friction terms and develop a well-balanced central-upwind scheme that is capable of exactly preserving its steady states. The scheme also preserves the positivity of the water depth. We test the designed scheme on a number of one-and two-dimensional examples that demonstrate robustness and high resolution of the proposed numerical approach. The data in the last numerical example correspond to the laboratory experiments reported in [L. Cea, M. Garrido, and J. Puertas, J. Hydrol, 382 (2010), pp. 88-102], designed to mimic the rain water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, we develop a special technique, which helps to achieve a remarkable agreement between the numerical and experimental results.

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normalΔt \leq {(normalΔt)}_{*} : = \frac{normalΔx}{2 a}, a = \max_{j} a_{j + \frac{1}{2}}^{+}, - a_{j + \frac{1}{2}}^{-},

where

a_{j + \frac{1}{2}}^{\pm}

{\overset{h}{false}}_{j}^{I} : = {\overset{w}{false}}_{j}^{I} - B_{j}

for all j provided

{\overset{h}{false}}_{j} (t) \geq 0

for all j .…”

Section: Methodsmentioning

confidence: 67%

Section: Methodsmentioning

confidence: 99%

\begin{matrix} w_{j}^{I} & = {\overset{true¯}{w}}_{j} (t) + Δt scriptF (1) {[\overset{true¯}{U} (t)]}_{j}, \\ q_{j}^{I} & = \frac{{\overset{true¯}{q}}_{j} (t) + Δt scriptF (2) {[\overset{true¯}{U} (t)]}_{j}}{1 - Δt G (\overset{true¯}{U} {(t)}_{j})}, \end{matrix}

B_{j, k} : = B (x_{j}, y_{k}) = \frac{1}{x y} B (x, y) d x d y = \frac{1}{4} (B_{j + \frac{1}{2}, k} + B_{j - \frac{1}{2}, k} + B_{j, k + \frac{1}{2}} + B_{j, k - \frac{1}{2}})

\begin{matrix} \frac{d}{dt} {\overset{w}{false}}_{j} & = F (1) {[\overset{bold-italicU}{false}]}_{j}, \\ \frac{d}{dt} {\overset{q}{false}}_{j} & = \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

Section: Time Integration Methodsmentioning

confidence: 99%

Section: Remark 35mentioning

confidence: 99%

See 3 more Smart Citations

Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

Chertock

Cui

Kurganov

et al. 2015

Numerical Methods in Fluids

View full text Add to dashboard Cite

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Recent Developments in the Field of Modified Patankar‐Runge‐Kutta‐methods

Izgin

Kopecz

Meister

2021

Proc Appl Math & Mech

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Modified Patankar‐Runge‐Kutta (MPRK) schemes are numerical one‐step methods for the solution of positive and conservative production‐destruction systems (PDS). They adapt explicit Runge‐Kutta schemes in a way to ensure positivity and conservation of the numerical approximation irrespective of the chosen time step size. Due to nonlinear relationships between the next and current iterate, the stability analysis for such schemes is lacking. In this work, we introduce a strategy to analyze the MPRK22(α)‐schemes in the case of positive and conservative PDS. Thereby, we point out that a usual stability analysis based on Dahlquist's equation is not possible in order to understand the properties of this class of schemes.

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A robust second-order surface reconstruction for shallow water flows with a discontinuous topography and a Manning friction

Dong

2020

Adv Comput Math

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Steady State and Sign Preserving Semi-Implicit Runge--Kutta Methods for ODEs with Stiff Damping Term

Cited by 41 publications

References 20 publications

Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

Recent Developments in the Field of Modified Patankar‐Runge‐Kutta‐methods

A robust second-order surface reconstruction for shallow water flows with a discontinuous topography and a Manning friction

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