High-Order, Finite-Volume Methods in Mapped Coordinates

β 1 i+2 ed +e❞ ❞ ❞ d′❞β i+❞ ❞ 1 2 ed ❞ ❞β i+ 1 2 ed −e❞ ❞ ❞ d′❞Fig. 1. Stencil for 〈∂Φ/∂ξ d 〉 1 The values of Φ at the cell centers marked by thei+2ed.open circles determine the β values given by (29) at the cell faces marked by thesolid circles.where the Φ i denote pointwise values of Φ at cell centers, we have∆β 1∂Φ| 1∂ξ = i+2 edi++ O(h 4 ), (30)d 2 ed h| 1∂ξ = 1 ∑()i+βd 2 ed h 3 1 i++ βd ′ ≠d 2 ed +e d′ 1 i+− 2β2 ed −e d′ 1 i++ O ( h 2) . (31)2 ed⊥,d ∂ΦFurthermore, we set( ) ∂ΦG ⊥,d0≡ 1 ()β∂ξ d i+ 1 h 2 1 i+− β2 ed 2 ed +e d′ 1 i+. (32)2 ed −e d′Next, for d ′ ≠ d, we have〈 〉 ∂Φ∂ξ d ′i+ 1 2 ed =( ∂Φ∂ξ d ′+ h2 ∂ 3 )Φ+ O ( h 4) . (33)24 ∂ξd 3 ′ i+ 1 2 ed10