MUSCL Scheme - Piecewise Parabolic Reconstruction

Piecewise Parabolic Reconstruction

It is possible to extend the idea of linear-extrapolation to higher order reconstruction, and an example is shown in the diagram opposite. However, for this case the left and right states are estimated by interpolation of a second-order, upwind biased, difference equation. This results in a parabolic reconstruction scheme that is third-order accurate in space.

We follow the approach of Kermani (Kermani, et al., 2003), and present a third-order upwind biased scheme, where the symbols and again represent scheme dependent functions (of the limited reconstructed cell edge variables). But for this case they are based upon parabolically reconstructed states, i.e.

$u^*_{i + \frac{1}{2}} = f \left( u^L_{i + \frac{1}{2}}, u^R_{i + \frac{1}{2}} \right), u^*_{i - \frac{1}{2}} = f \left( u^L_{i - \frac{1}{2}}, u^R_{i - \frac{1}{2}} \right),$

and

$u^L_{i + \frac{1}{2}} = u_{i} + \frac{\phi \left( r_{i} \right)}{4} \left[ \left( 1 - \kappa \right) \delta u_{i - \frac{1}{2} } + \left( 1 + \kappa \right) \delta u_{i + \frac{1}{2} } \right],$

$u^R_{i + \frac{1}{2}} = u_{i+1} - \frac{\phi \left( r_{i+1} \right)}{4} \left[ \left( 1 - \kappa \right) \delta u_{i + \frac{3}{2} } + \left( 1 + \kappa \right) \delta u_{i + \frac{1}{2} } \right],$

$u^L_{i - \frac{1}{2}} = u_{i-1} + \frac{\phi \left( r_{i-1} \right)}{4} \left[ \left( 1 - \kappa \right) \delta u_{i - \frac{3}{2}} + \left( 1 + \kappa \right) \delta u_{i - \frac{1}{2} } \right],$

$u^R_{i - \frac{1}{2}} = u_{i} - \frac{\phi \left( r_{i} \right)}{4} \left[ \left( 1 - \kappa \right) \delta u_{i + \frac{1}{2} } + \left( 1 + \kappa \right) \delta u_{i - \frac{1}{2} } \right].$

Where = 1/3 and,

$\delta u_{i + \frac{1}{2} } = \left( u_{i+1} - u_{i} \right), \delta u_{i - \frac{1}{2} } = \left( u_{i} - u_{i-1} \right),$

$\delta u_{i + \frac{3}{2} } = \left( u_{i+2} - u_{i+1} \right), \delta u_{i - \frac{3}{2} } = \left( u_{i-1} - u_{i-2} \right),$

and the limiter function, is the same as above.

Parabolic reconstruction is straight forward to implement and can be used with the Kurganov and Tadmor scheme in lieu of the linear extrapolation shown above. This has the effect of raising the spatial solution of the KT scheme to 3rd order. It performs well when solving the Euler equations, see below. This increase in spatial order has certain advantages over 2nd order schemes for smooth solutions, however, for shocks it is more dissipative - compare diagram opposite with above solution obtained using the KT algorithm with linear extrapolation and Superbee limiter. This simulation was carried out on a mesh of 200 cells using the same KT algorithm but with parabolic reconstruction. Time integration was by RK-4, and the alternative form of van Albada limiter, was used to avoid spurious oscillations.

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