Central Runge--Kutta Schemes for Conservation Laws

Pareschi, Lorenzo; Puppo, Gabriella; Russo, Giovanni

doi:10.1137/s1064827503420696

Cited by 32 publications

(51 citation statements)

References 38 publications

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“…In our experience, fourth order schemes seem to be particularly effective whenever high resolution is called for [12]. For this reason we will compare four different fourth order schemes.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The piecewise parabolic WENO reconstruction is fifth order accurate for the evaluation of point values for unstaggered schemes, while it is fourth order accurate for central schemes based on staggered grids, unless a splitting of the weights in their positive and negative parts is used, see [12].…”

Section: Weno Reconstructionmentioning

confidence: 99%

“…Since the reconstruction polynomials are smooth at the grid points x j , these quantities can be computed using the differential form of the PDE, as in CRK schemes [12]:…”

Section: Staggered Finite Difference Schemesmentioning

confidence: 99%

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Staggered Finite Difference Schemes for Conservation Laws

Puppo

Russo

2006

J Sci Comput

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Here we show how to construct finite-difference shock-capturing central schemes on staggered grids. Staggered schemes may have better resolution of the corresponding unstaggered schemes of the same order. They are based on high order non oscillatory reconstruction (ENO or WENO), and a suitable ODE solver for the computation of the integral of the flux. Although they suffer a more severe stability restriction, they do not require a numerical flux function. A comparison between central finite volume and finite difference, on staggered and non staggered grids, is reported.

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“…In our experience, fourth order schemes seem to be particularly effective whenever high resolution is called for [12]. For this reason we will compare four different fourth order schemes.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Weno Reconstructionmentioning

confidence: 99%

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Staggered Finite Difference Schemes for Conservation Laws

Puppo

Russo

2006

J Sci Comput

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“…In order to evaluate the time integrals of the source term (19) and the time flux integrals in (7) we have to predict the point-values of the solution at two intermediate states:û n+β k i,j ≡ u(x i , y j , t n + β k ∆t), k = 0, 1. The prediction of these intermediate values at times t n+β 0 and t n+β 1 is obtained by means of a Runge-Kutta scheme coupled with the natural continuous extension (NCE) [5]:û…”

Section: Reconstruction Of Runge-kutta Fluxesmentioning

confidence: 99%

“…Central schemes of Bianco et al [5], Levy et al [14], Pareschi et al [19] or Balaguer-Beser [2] are particularly useful when the eigensystem of the equations is very complex or even unknown. Balaguer and Conde [1] compare the accuracy of upwind and central schemes to solve scalar conservation laws.…”

Section: Introductionmentioning

confidence: 99%

A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

Capilla

Balaguer-Beser

2013

Journal of Computational and Applied Mathematics

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This paper is concerned with the development of high-order well-balanced central schemes to solve the shallow water equations in two spatial dimensions. A Runge-Kutta scheme with a natural continuous extension is applied for time discretization. A Gaussian quadrature rule is used to evaluate time integrals and a three-degree polynomial which calculates point-values from cell averages or flux values by avoiding the increase in the number of solution extrema at the interior of each cell is used as reconstruction operator. That polynomial also guarantees that the number of extrema does not exceed the initial number of extrema and thus it avoids spurious numerical oscillations in the computed solution. A new procedure has been defined to evaluate the flux integrals and to approach the 2D source term integrals in order to verify the exact C-property, using the water surface elevation instead of the water depth as a variable. Numerical experiments have confirmed the high-resolution properties of our numerical scheme in 2D test problems. The well-balanced property of the resulting scheme has also been investigated.

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A new reconstruction procedure in central schemes for hyperbolic conservation laws

Balaguer-Beser

2011

Numerical Meth Engineering

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SUMMARYThis paper presents a new point value reconstruction algorithm based on average values or flux values for central Runge-Kutta schemes in the resolution of hyperbolic conservation laws. This reconstruction employs a fourth-order accurate approximation of point values of the solution at the two extrema and at the mid-point of each cell. These point values are modified in order to enforce monotonicity and shape preserving properties. This correction has been applied essentially in the cells close to the maxima and minima of the solution and in these cases, it has been proven that the reconstruction is fourth-order accurate. In the cells with a maximum or minimum of the solution, a correction has also been applied to such point values with the aim of ensuring that the resulting numerical solution has a non-oscillatory behavior. Several standard one-and two-dimensional test cases are used to verify high-order accuracy, non-oscillatory behavior and high-resolution properties for smooth and discontinuous solutions, and also in their componentwise extension to the Euler gas dynamics equations.

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Central Runge--Kutta Schemes for Conservation Laws

Cited by 32 publications

References 38 publications

Staggered Finite Difference Schemes for Conservation Laws

Staggered Finite Difference Schemes for Conservation Laws

A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

A new reconstruction procedure in central schemes for hyperbolic conservation laws

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