Abstract
We present an approach to designing arbitrarily high-order finite-volume spatial discretizations on locally-rectangular grids. It is based on the use of a simple class of high-order quadratures for computing the average of fluxes over faces. This approach has the advantage of being a variation on widely-used second-order methods, so that the prior experience in engineering those methods carries over in the higher-order case. Among the issues discussed are the basic design principles for uniform grids, the extension to locally-refined nest grid hierarchies, and the treatment of complex geometries using mapped grids, multiblock grids, and cut-cell representations.
Highlights
It is often the case that one wants to compute solutions to partial differential equations containing terms of the form ∇ · F, F = (F 1, . . . , F D), F d = F d(x)
For the embedded boundary approach, we have to define methods for computing high-order approximations to the average of ∇ · F over irregular control volumes generated where Cartesian control volumes intersect the irregular domain, as well as for faces are sufficiently close to the boundary to require an irregular stencil
The principal focus of the discussion here has been to lay out an approach for extending classical finite-volume methods for (1), (2) to obtain discretization methods of any order of accuracy in the truncation error
Summary
We describe one specific approach to designing arbitrarily highorder finite-volume spatial discretizations on locally-rectangular grids It is based on the use of a simple class of high-order quadratures for computing the average of fluxes over faces. But straightforward, to apply the approach given above to compute the derivatives in the right hand side to the necessary order - the process can be automated using a symbolic algebra package These formulas apply well to transform between face averages and point values at the centers of faces, except that the sums over the derivatives in the correction terms exclude the derivatives with respect to the direction normal to the face over which one is averaging.
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