High-Order, Finite-Volume Methods in Mapped Coordinates

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where we adopt the pointwise notation q i±1 ≡ q(ξ 2 ed i± 1 ed). Thus, the faceaveragedflux is expressed in terms of pointwise values of v, u, and their2derivatives at the center of the face i ± 1 2 ed .The expansion of the integral (63) also gives pointwise values expressed interms of face-averaged values. Specifically, for pointwise values of u and v,one can writev i±1 = 〈v〉 h22 ed i± 1 −ed2 24u i±1 = 〈u〉 h22 ed i± 1 −ed2 24∑d ′ ≠d∑d ′ ≠d∂ 2 v∂ξ 2 d ′ ∣ ∣∣∣∣ξ=ξi±12e d + O ( h 4) ,∂ 2 u∂ξ 2 d ′ ∣ ∣∣∣∣ξ=ξi±12e d + O ( h 4) .(65a)(65b)Thus, the average interface flux is〈F〉 i±1 = 〈v〉 2 ed i± 1 ed〈u〉 h2i± 1 +ed2 2 12(∑ ∂vd ′ ≠d)∂u∂ξ d ′ ∂ξ d ′ξ=ξ i± 12e d + O ( h 4) , (66)written in terms of the face-averages 〈u〉 i±1 and 〈v〉 2 ed i± 1 and the pointwiseed2values of the gradients of u and v at the center of the face.In a finite-volume scheme, one works with cell-averaged values. Through theuse of primitive functions [16], one can construct face-averages directly interms of cell averages; at fourth-order on a uniform grid, this yields thecentrally-differenced expression〈q〉 i+12 ed = 7 12 (¯q i + ¯q i+e d) − 1 12 (¯q i+2e d + ¯q i−e d) + O( h 4) . (67)To approximate the pointwise transverse gradients, we first note that an O(h 2 )approximation is sufficient. In the d ′ -th direction, a suitable centrally differencedapproximation is∂q= 1 (¯qi+e d∂ξ d ′ ∣ ξ=ξi±4h′ + ¯q i±e d +e − ¯q d′ i−e − ¯q ) ( )d′ i±e d −e + O h2. (68)d′12e dExpressions (67) and (68) provide the approximations necessary to evaluatethe face-averaged fluxes (66) to fourth-order given the cell averages of u andv on the computational grid.To obtain the average ū i from the average (uJ) i, we again appeal to Taylorseries expansion of the integrand to express the average of products as theproduct of averages:(uJ) i= ū i ¯Ji + h212 (∇ ξu · ∇ ξ J) i+ O ( h 4) . (69)20
3Runge Kutta Stability Domain21Im z0−1−2−3−2.5 −2 −1.5 −1 −0.5 0Re zFig. 6. Stability region of the classical Runge-Kutta scheme. The approximation bya semi-ellipse (77) is shown in blue.Thus,ū i ≡ h −D ∫V iu(x(ξ))dξ = ( [−1 ¯J) (uJ)i i− h212 ∇ ξu · ∇ ξ J + O ( h 4)] . (70)We require at least second-order approximations of the gradients and choosethe central differences⎡(∇ ξ u) d i= 1 ⎣ (uJ) i+e d2h ¯J i+e d− (uJ) i−e d¯J i−e d⎤⎦ + O ( h 2) (71)in each direction d. This choice is freestream preserving; the difference evaluatesto zero (within roundoff) provided that the averages are initialized suchthat, for constant u, uJ i ≡ u ¯J i .4.2 Time DiscretizationAs in [7], we discretize the semi-discrete system of ordinary differential equations(61) using the explicit, four-stage, fourth-order classical Runge-Kuttascheme [17]. Consider the variable-coefficient problemdydt= A(x)y, (72)21
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 13 and 14: ✠Dirichlet boundaryDCBAFig. 3. Lo
Page 15 and 16: 10.80.60.40.20−0.2−0.4−0.6−
Page 17: 10 −2 N10 −310 −410 −510
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

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