A Third-Order Unconditionally Positivity-Preserving Scheme for Production–Destruction Equations with Applications to Non-equilibrium Flows

Huang, Juntao; Zhao, Weifeng; Shu, Chi‐Wang

doi:10.1007/s10915-018-0881-9

Cited by 36 publications

(66 citation statements)

References 22 publications

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“…While our method applies to general IMEX RK schemes, we demonstrate its effectiveness with a specific third-order scheme in [7] by fixing the CFL number as 0.8. The third-order accuracy of the method is verified for both 1D and 2D cases, and the numerical results of the 2D reactive Euler equations are comparable to those obtained by the third-order positivity-preserving schemes with reflective boundary conditions [17].…”

Section: Introductionmentioning

confidence: 65%

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

Zhao,

Huang

2019

Preprint

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In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme.We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff procedure. The former preserves the accuracy of the RK schemes and the latter guarantees the stability. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [12], which could not be derived for the IMEX schemes. In addition, the intermediate boundary conditions are only available for explicit RK schemes up to third order while our method applies to arbitrary order IMEX and explicit RK schemes. Moreover, the present boundary treatment method may be adapted to IMEX RK schemes solving many other partial differential equations. For a specific third-order IMEX scheme, we demonstrate the good stability and third-order accuracy of our boundary treatment through both 1D examples and 2D reactive Euler equations.

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Section: Introductionmentioning

confidence: 65%

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

Zhao,

Huang

2019

Preprint

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“…Applications for PDS are for example the biological and/or chemical reactions such as algal bloom [12]. Also parts (or all) of the semi discretization of hyperbolic conservation/balance laws can be interpreted in such PDS system as described in [32,33,41] and also later in this work. The calculated solutions are often describing physical quantities that enjoy some properties, for instance concentrations of chemicals or water height in the context of SWE should be nonnegative.…”

Section: Patankar Methods For Production-destruction Systemsmentioning

confidence: 99%

“…Based on this technique, extensions to second and third order modified Patankar Runge-Kutta (MPRK) methods have been made by several researchers in such context, cf. [32,33,35,36]. Also the semi implicit RK methods proposed in [17] can be interpreted as Patankar methods as they weight only the destruction terms [62].…”

Section: Patankar Methods For Production-destruction Systemsmentioning

confidence: 99%

“…Then, to ensure the positivity of the water height, the modified Patankar Deferred Correction (mPDeC) method is used for the time-integration of this variable. Even if (modified) Patankar (mP) methods have been already used inside a numerical method for fluid simulations [32,33,42], those mP schemes have been based on extensions of classical RK methods and they are of maximum order three 1 . By applying the modified Patankar trick inside the Deferred Correction (DeC) framework, the authors of [48] were able to construct a conservative, arbitrarily high-order and positivity preserving method for production-destruction systems (PDS) of ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

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An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

Ciallella¹,

Lorenzo²,

Öffner³

et al. 2021

Preprint

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In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.

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“…We believe that common positive preserving techniques like those developed in [30,31,32] could potentially be a solution to handle this instability issue.…”

Section: Numerical Testsmentioning

confidence: 99%

A Numerical Scheme for Wave Turbulence: 3-Wave Kinetic Equations

Walton¹,

Tran²

2021

Preprint

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We introduce a finite volume scheme to solve isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions verify the energy cascade phenomenon. Up to our knowledge, this is the first numerical scheme that could capture the long time asymptotic behavior of solutions to isotropic 3-wave kinetic equations, where the energy cascade can be observed. Our numerical energy cascade rates are in good agreement with the theoretical one obtained by . Our finite volume algorithm relies on a new identity, that allows one to reduce the number of terms needed to be approximated in the collision operators.

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A Third-Order Unconditionally Positivity-Preserving Scheme for Production–Destruction Equations with Applications to Non-equilibrium Flows

Cited by 36 publications

References 22 publications

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

A Numerical Scheme for Wave Turbulence: 3-Wave Kinetic Equations

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