High-Order, Finite-Volume Methods in Mapped Coordinates

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we furthermore setG ⊥,d0( ∂Φ≡)i+ 1 (1 h 22 ed∂ξ d ′)γ d′ ,(4),+i+ 1 − γ d′ ,(4),−2 ed i+ 1 . (40)2 edThe stencil entries given by (39) yield fourth-order accurate first derivatives inthe d ′ direction at the face centers i+ 1 2 ed ±e d′ . We employ these non-centeredformulas to ensure that the resulting stencil is confined to block of cells atmost 5 cells wide centered on the cell in which the flux divergence average isbeing computed.3.2 Boundary conditionsFor boundaries upon which a Dirichlet boundary condition is posed, the faceaverages 〈∂Φ/∂ξ d ′〉 1 i+used in (25) on faces contained in such boundaries caned2be computed using modified discretizations that incorporate the prescribedboundary values. Suppose that the cell face with center at i + 1 2 ed is onesuch face, such as the face centered on the point labeled A in Figure 3. Theaverages 〈∂Φ/∂ξ d ′〉 1 for the transverse coordinates d′ i+ ≠ d can presumably2 edbe computed directly from prescribed Dirichlet data to fourth-order accuracy.For the normal direction, the stencil describing 〈∂Φ/∂ξ d 〉 1 i+can be modifieded2by replacing the definition (29) byβ (4)i+ 1 ≡ 1 (2816Φi+ 12 ed 840− 3675Φ 2 ed i+ 1225Φ i−e d − 441Φ i−2e d + 75Φ i−3e d),(41)where Φ 1 i+is the prescribed boundary value at the center of the cell face.ed2Although this formula yields a fourth order accurate approximation of the normalderivative, it results in a stencil extending beyond the 5-cell-wide blockcentered about the cell upon which the discrete divergence is being computed.To avoid using a larger stencil at the boundary than in the interior, we take advantageof the opportunity to reduce the discretization order at the boundarywhile still maintaining fourth-order accuracy overall due to elliptic regularity.In particular, instead of (41) we defineβ (4)i+ 1 ≡ 1 ()184Φ 12 ed 60− 225Φ i+2 ed i + 50Φ i−e d − 9Φ i−2e d . (42)The same issue affects the normal and transverse derivatives on interior facesparallel to the boundary exactly one cell away, such as the face centered onthe point labeled B in Figure 3, i.e., the use of a non-centered fourth-order12
✠Dirichlet boundaryDCBAFig. 3. Locations at or near a Dirichlet boundary requiring stencil modifications(see text).discretization leads to a stencil that is not contained in a 5×5 block. However,we may use the same interpolating cubic polynomial used in finding (42) toobtainβ (4)i+ 1 ≡ 1 ()−8Φ 12 ed 60+ 75Φ i+2 ed i − 70Φ i−e d + 3Φ i−2e d . (43)Similarly, (34) is replaced byα i+12 ed ≡ 1 20()−4Φ 1 + 15Φ i+2 ed i + 10Φ i−e d − Φ i−2e d . (44)Next, consider cell faces adjacent and normal to a Dirichlet boundary, suchas the face centered on the point labeled C in Figure 3. For the calculation ofthe average of the normal derivative (d = d ′ ), the calculation of the transverseLaplacian (31) using centered differences can be shifted one cell away from theboundary with no loss of the required second-order accuracy. A non-centered,second-order accurate formula replaces (32):( ) ∂ΦG ⊥,d0≡ 1 ()3β∂ξ d i+ 1 2h 2 1 − 4β i+2 ed 2 ed i+ 1 + β2 ed −e d′ 1 i+. (45)2 ed −2e d′For the average of the transverse derivative (d ≠ d ′ ), we replace (35) and (36)13
Page 1 and 2: High-Order, Finite-Volume Methods i
Page 3 and 4: the exact enforcement of the metric
Page 5: If we define control volumes in phy
Page 8 and 9: Alternative expressions to (17) are
Page 10: β 1 i+2 ed +e❞ ❞ ❞ d′❞β
Page 15 and 16: 10.80.60.40.20−0.2−0.4−0.6−
Page 17: 10 −2 N10 −310 −410 −510
Page 21 and 22: 3Runge Kutta Stability Domain21Im z
Page 23 and 24: Magnitude of Amplification FactorRe
Page 26 and 27: L ∞ Error Norm for Base SchemeN c
Page 28 and 29: 10 0 Uniform Advection of Sinusoida
Page 30 and 31: Note that, in computational space,
Page 32 and 33: 10 0 Uniform Advection of Compact D
Page 34 and 35: and/or face-centered values. Among

High-Order, Finite-Volume Methods in Mapped Coordinates

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