High-Order, Finite-Volume Methods in Mapped Coordinates

Following (19) and (20), we therefore obtain∫X(V i )∇ x · Fdx = h 23∑∑d=1 ±=+,−±F di± 1 2 ed + O ( h 4) , (24)where, using face-centered pointwise values of F in the transverse gradients,〈 〉3∑F di+ 1 ≡ 〈F d〉 1 = ∂Φ2 ed i+˜D dd ′2 ed ∂ξd ′ =1d ′i+ 1 ⎡2 ed3∑=〈˜D 〈 〉⎣ ∂Φdd ′〉i+ 1 d ′ =12 ed ∂ξ d ′i+ 1 2 ed+ h212 G⊥,d 0(〈˜D dd ′〉)i+ 1 · G⊥,d 0ed2( ∂Φ∂ξ d ′)i+ 1 2 ed ⎤⎦(25)where˜D ≡ (˜D dd ′)≡ J −1 N T DN. (26)Face averages 〈˜D dd ′〉can be computed to fourth order in terms of face averagesof the entries of the factor matrices N T , D and J −1 N using the productformula (17). Computing the second-order accurate transverse gradientsG ⊥,d0(〈˜D dd ′〉)i+ 1 ≡ 1 (〉 〈˜D dd ′2 ed hi+ 1 − 〈˜D )〉dd2 ed +e d′ ′i+ 1 , (27)2 ed −e d′it then remains to specify the discretization of the averages 〈∂Φ/∂ξ d ′〉 1 i+2 edand transverse gradient G ⊥,d0 (∂Φ/∂ξ d ′) 1 i+ ed. 23.1 Discretization of 〈∂Φ/∂ξ d ′〉 1 i+and G⊥,d 0 (∂Φ/∂ξed d ′) 1 i+2 2 edFirst consider the case where d ′ = d. We have〈 ∂Φ∂ξ d〉i+ 1 2 ed =( )∂Φ+ h2 ∂Φ∆⊥,d + O ( h 4) , (28)∂ξ d 24 ∂ξ d i+ 1 2 edwhere ∆ ⊥,d is the Laplacian in the directions transverse to the d-th direction.Definingβ i+12 ed ≡ 1 24 [27 (Φ i+e d − Φ i) − (Φ i+2e d − Φ i−e d)] , (29)9