High-Order, Finite-Volume Methods in Mapped Coordinates

ule applied to determinants, we have∑∂(det ( (∇ ξ X) (X|d ′ )(s|e d ) ))d ′ ≠d∂ξ d ′= ∑ det ( (∇ ξ X) ( ∂X |d ′ )(s|e d ) )∂ξd ′ ≠dd ′+ ∑ ∑det ( (∇ ξ X) (X|d ′ ∂ 2 X)( |d ′′ )(s|e d ) ) .∂ξd ′ ≠d d ′′ ≠d,d ′ d ′∂ξ d ′′(15)Each summand in the first (single) sum is just Nd, s so it suffices to show thatthe second (double) sum vanishes. However, for a given d 1 , d 2 , d 1 ≠ d 2 , summandsin the double sum involving the mixed second partial2 X ∂∂ξ d1 ∂ξ d2appearexactly twice, differing from one another only by the exchange of the d 1 and d 2columns. By the antisymmetry of the determinant under column exchanges,the two summands cancel, and hence the entire second sum vanishes. Finally,we need to show the antisymmetry condition Nd,d s = −N s ′ d ,d. The following is′a consequence of linearity of the determinant as a function of the d ′ column,plus the identity det(A(e p |q)) = det(A(q|e p )):det ( (∇ ξ X) (X|d ′ )(s|e d ) ) = ∑ s ′ ≠sX s ′ det ( (∇ ξ X) (s|e d )(s ′ |e d′ ) ) . (16)The right-hand side of (16) is manifestly antisymmetric in d, d ′ .2.2 Fourth-order mapped-grid finite-volume discretizationFollowing these ideas, we can specify the information required for a fourthorderaccurate finite-volume discretization. Using a Taylor series, the integralson the cell faces A ± d can be approximated using the following formula for theaverage of a product in terms of fourth-order accurate averages of each factor:〈fg〉 i+12 ed =〈f〉 i+1 〈g〉2 ed i+1 + h22 ed 12 G⊥,d 0( ) ( )〈f〉i+ 1 · G⊥,d2 ed 0 〈g〉i+ 1 + O(h 4 ).2 ed(17)Here, G ⊥,d0 is the second-order accurate central difference approximation tothe components of the gradient operator orthogonal to the d-th direction:G ⊥,d0 ≈ ∇ ξ − e d ∂∂ξ d, and the operator 〈·〉 i+1 denotes a fourth-order accurateed2average over the face centered at i + 1 2 ed :〈q〉 i+1 = 1 ∫2 ed h D−1A dq(ξ)dA ξ + O(h 4 ). (18)7