The Picard Integral Formulation of Weighted Essentially Nonoscillatory Schemes

Christlieb, Andrew; Guclu, Yaman; Seal, David C.

doi:10.1137/140959936

Cited by 11 publications

(24 citation statements)

References 53 publications

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“…We now briefly review the Taylor-discretization Picard integral formulation weighted essentially non-oscillatory (PIF-WENO) method [23]. This method applies to generic hyperbolic conservation laws in arbitrary dimensions, of which the ideal MHD equation is an example.…”

Section: The Taylor-discretization Pif-weno Methodsmentioning

confidence: 99%

A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

Christlieb

Feng

Seal

et al. 2016

Journal of Computational Physics

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We propose a high-order finite difference weighted ENO (WENO) method for the ideal magnetohydrodynamics (MHD) equations. The proposed method is single-stage (i.e., it has no internal stages to store), single-step (i.e., it has no time history that needs to be stored), maintains a discrete divergence-free condition on the magnetic field, and has the capacity to preserve the positivity of the density and pressure. To accomplish this, we use a Taylor discretization of the Picard integral formulation (PIF) of the finite difference WENO method proposed in [SINUM, 53 (2015[SINUM, 53 ( ), pp. 1833[SINUM, 53 ( -1856, where the focus is on a high-order discretization of the fluxes (as opposed to the conserved variables). We use the version where fluxes are expanded to third-order accuracy in time, and for the fluid variables space is discretized using the classical fifth-order finite difference WENO discretization. We use constrained transport in order to obtain divergence-free magnetic fields, which means that we simultaneously evolve the magnetohydrodynamic (that has an evolution equation for the magnetic field) and magnetic potential equations alongside each other, and set the magnetic field to be the (discrete) curl of the magnetic potential after each time step. In this work, we compute these derivatives to fourth-order accuracy. In order to retain a single-stage, single-step method, we develop a novel Lax-Wendroff discretization for the evolution of the magnetic potential, where we start with technology used for Hamilton-Jacobi equations in order to construct a non-oscillatory magnetic field. The end result is an algorithm that is similar to our previous work [JCP, 268, (2014), pp. 302-325], but this time the time stepping is replaced through a Taylor method with the addition of a positivitypreserving limiter. Finally, positivity preservation is realized by introducing a parameterized flux limiter that considers a linear combination of high and low-order numerical fluxes. The choice of the free parameter is then given in such a way that the fluxes are limited towards the low-order solver until positivity is attained. Given the lack of additional degrees of freedom in the system, this positivity limiter lacks energy conservation where the limiter turns on. However, this ingredient can be dropped for problems where the pressure does not become negative. We present two and three dimensional numerical results for several standard test problems including a smooth Alfvén wave (to verify formal order of accuracy), shock tube problems (to test the shock-capturing ability of the scheme), Orszag-Tang, and cloud shock interactions. These results assert the robustness and verify the high-order of accuracy of the proposed scheme.

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Section: The Taylor-discretization Pif-weno Methodsmentioning

confidence: 99%

A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

Christlieb

Feng

Seal

et al. 2016

Journal of Computational Physics

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“…The Euler equations are an example of a set of equations from this class of problems. Formal integration of (7) in time over t ∈ [t n ,t n+1 ] defines the 2D Picard integral formulation [16] as…”

Section: A Single-stage Single-step Finite Difference Weno Methodsmentioning

confidence: 99%

“…The purpose of this work is to define a single-stage, single-step finite difference WENO method that is provably positivity-preserving for the compressible Euler equations. Of the various finite difference schemes constructed from the Picard integral formulation [16], we begin with the Taylor discretization, and then apply recently developed flux limiters [13,57] in order to retain positivity of the solution. One advantage of the chosen limiter is that positivity is preserved without introducing additional time step restrictions, however, our primary contribution is that the present method is the first scheme to simultaneously be high-order, single-stage, single-step and have provable positivity preservation.…”

Section: Introductionmentioning

confidence: 99%

An Explicit High-Order Single-Stage Single-Step Positivity-Preserving Finite Difference WENO Method for the Compressible Euler Equations

Seal

Tang

et al. 2015

J Sci Comput

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In this work we construct a high-order, single-stage, single-step positivity-preserving method for the compressible Euler equations. Space is discretized with the finite difference weighted essentially non-oscillatory (WENO) method. Time is discretized through a LaxWendroff procedure that is constructed from the Picard integral formulation (PIF) of the partial differential equation. The method can be viewed as a modified flux approach, where a linear combination of a low-and high-order flux defines the numerical flux used for a single-step update. The coefficients of the linear combination are constructed by solving a simple optimization problem at each time step. The high-order flux itself is constructed through the use of Taylor series and the Cauchy-Kowalewski procedure that incorporates higher-order terms. Numerical results in one-and two-dimensions are presented.

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“…Now, we consider a solution u(x, t) to the heat equation (2.1), that for simplicity we take to be infinitely smooth. We perform a Taylor expansion on u(x, t + ∆t), and then use the Cauchy-Kovalevskaya procedure [9,40] to exchange temporal and spatial derivatives to yield…”

Section: Homogeneous Solutionmentioning

confidence: 99%

Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution

Causley

Cho

Christlieb³

et al. 2016

SIAM J. Numer. Anal.

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We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N ) convolution. This fundamental solver has arbitrary order of accuracy in space, and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions.

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The Picard Integral Formulation of Weighted Essentially Nonoscillatory Schemes

Cited by 11 publications

References 53 publications

A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

An Explicit High-Order Single-Stage Single-Step Positivity-Preserving Finite Difference WENO Method for the Compressible Euler Equations

Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution

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