AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

Transcription

1 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 59 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS The Finite Volume Method These slides are partially based on the recommended textbook: Culbert B. Laney. Computational Gas Dynamics, CAMBRIDGE UNIVERSITY PRESS, ISBN / 59

2 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 59 Outline 1 Conservative Finite Volume Methods in One Dimension 2 Introduction to Reconstruction-Evolution Methods 3 First-Order Upwind Reconstruction-Evolution Methods Scalar Conservation Laws Euler Equations Roe s First-Order Upwind Method for the Euler Equations 4 Introduction to Second- & Higher-Order Reconstruction-Evolution Methods 5 The MUSCL/TVD Method The Method of Lines Flux Splitting Second-Order Approximation Van Leer s Slope Limiter 6 Steger-Warming Flux Vector Splitting Method for the Euler Equations 7 Multidimensional Extensions 2 / 59

3 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 3 / 59 Note: The material covered in this chapter equally applies to scalar conservation laws and the Euler equations, in one and multiple dimensions. To keep matters as simple as possible however, this material is presented primarily in one dimension then briefly extended to multiple dimensions. 3 / 59

4 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 4 / 59 Conservative Finite Volume Methods in One Dimension Recall that the integral form of a conservation law can be written as xi+1/2 x i 1/2 [ux, t n+1 ) ux, t n )] dx = t n+1 t n [f ux i+1/2, t) ) f ux i 1/2, t) ) ] dt This leads to the following numerical conservation form where xi+1/2 t ū t )n i = λˆf n i+1/2 ˆf n i 1/2 ) 1) t n+1 ūi n 1 ux, t n )dx, ˆf i+1/2 n x 1 f ux i+1/2, t) ) dt x i 1/2 t t n 2) and the rest of the notation is the same as in the previous Chapter 4 / 59

5 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 5 / 59 Conservative Finite Volume Methods in One Dimension ūi n is the spatial cell-integral average value of u at time t n that is, the average value of u in the cell [x i 1/2, x i+1/2 ] at time t n ˆf i+1/2 n is the time-integral average of f at the point x i+1/2 Recall that 1 xi+1/2 gx)dx = gx i ) + O x 2 ) x x i 1/2 It follows that if ui n is replaced by ūi n, ALL concepts and results presented in the previous Chapter that is, for the finite difference method equally apply for the finite volume method presented in this Chapter ūi n = ui n + O x 2 ) second-order accuracy in space is not affected by identifying ūi n with ui n and vice-versa there is no need to distinguish between finite volume and finite difference when the order of spatial accuracy is less or equal to 2 the bar notation is often dropped in the remainder of this chapter, particularly when this should not create any confusion 5 / 59

6 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 6 / 59 Introduction to Reconstruction-Evolution Methods Two-step finite volume design approach spatial reconstruction reconstruct ux, t n ) without necessarily accounting for the upwind direction this step differentiates the finite volume method which forms u then f u), from the finite difference method which forms directly f u) temporal evolution approximate ux i+1/2, t) for t n t t n+1 then evaluate ˆf n i+1/2 = 1 t t n+1 t n f ux i+1/2, t) ) dt any reasonable approximation based on waves and characteristics naturally introduces the minimal amount of upwinding required by the CFL condition Reconstruction-evolution methods are sometimes called Godunov-type methods or MUSCL-type methods 6 / 59

7 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 7 / 59 Introduction to Reconstruction-Evolution Methods 7 / 59

8 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 8 / 59 Introduction to Reconstruction-Evolution Methods Example: first-order reconstruction-evolution method for the linear advection equation piecewise constant reconstruction in the cell [x i 1/2, x i+1/2 ]: ux, t n ) p r x) = ūi n for the linear advection equation, ux, t) = ux at t n ), t n ) and therefore ux i+1/2, t) p e,i+1/2 t) = p r xi+1/2 at t n ) ) { ūn = i for 0 λa 1 ūi+1 n for 1 λa 0 then ˆf n i+1/2 = = t n+1 1 f p e,i+1/2 t) ) dt = 1 t t n t { aū n i for 0 λa 1 aūi+1 n for 1 λa 0 t n+1 t n ap e,i+1/2 t)dt 8 / 59

9 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 9 / 59 Introduction to Reconstruction-Evolution Methods What if the previous example was about a nonlinear conservation law instead of the linear advection equation? piecewise constant reconstruction gives rise to a Riemann problem at each cell edge, and the Riemann problem has an exact solution at x/t = 0 an approximate Riemann solver can be used instead of the true Riemann solver without changing the numerical solution higher-order reconstruction the jump at each cell edge in the higher-order piecewise polynomial reconstruction gives rise to a problem that lacks a known exact solution the exact solution can be approximated using Riemann solvers 9 / 59

10 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 10 / 59 Introduction to Reconstruction-Evolution Methods 10 / 59

11 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 11 / 59 First-Order Upwind Reconstruction-Evolution Methods Scalar Conservation Laws Suppose that the reconstruction is piecewise constant: Then, each cell edge gives rise to a Riemann problem The exact evolution of the piecewise constant reconstruction yields where ˆf n i+1/2 = u n+1 i = u n i λˆf n i+1/2 ˆf n i 1/2 ) 1 t t n+1 t n f u RIEMANN x i+1/2, t) ) dt Since the solution of the Riemann problem is self-similar a function of x x i+1/2 )/t) ), u RIEMANN x i+1/2, t) is constant for all time since x i+1/2 x i+1/2 )/t = 0 = ˆf n i+1/2 = f u RIEMANN x i+1/2, t) ) = f u RIEMANN 0)) ) 3) where f u RIEMANN x i+1/2, t) ) is computed using any exact or approximate Riemann solver and t is any arbitrary time t > t n Many first-order upwind methods for scalar conservation laws are based on 3) 11 / 59

12 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 12 / 59 First-Order Upwind Reconstruction-Evolution Methods Scalar Conservation Laws Upwind methods based on ˆf i+1/2 n = f u RIEMANN x i+1/2, t) ) equation 3) assumes that the waves from different cell edges do not interact or at least that any interaction does not affect the solution at the cell edges) λau) 1 waves originating at one cell edge cannot interact 2 with those originating from any other cell edge λau) 1 they can, but the interactions cannot reach the cell edges in one time-step upwind CFL condition: λau) 1 conservative, consistent, converge when x 0 and t 0 and the CFL condition is satisfied explicit, finite volume linearly stable provided the CFL condition is satisfied satisfy all nonlinear stability conditions discussed in the previous Chapter formally first-order accurate in time and space except possibly at sonic points) 12 / 59

13 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 13 / 59 First-Order Upwind Reconstruction-Evolution Methods Scalar Conservation Laws Upwind methods based on ˆf n i+1/2 = f u RIEMANN x i+1/2, t) ) continue) if λau) 1, waves can interact only if there is a compressive sonic sonic au) = 0, compressive wave direction switches from right to left) point inside the cell 1 waves at the left cell edge are right-running, counterparts at the right cell edge are left-running if the wave speeds are always positive, all waves originating from x i+1/2 are right-running ˆf i+1/2 n = f u RIEMANN x i+1/2, t) ) = f ui n) FTBS if the wave speeds are always negative, all waves originating from x i+1/2 are left-running ˆf i+1/2 n = f u RIEMANN x i+1/2, t) ) = f ui+1 n ) FTFS the above is true for the exact Riemann solver or any reasonable approximate Riemann solver all first-order upwind methods based on ˆf i+1/2 n = f u RIEMANN x i+1/2, t) ) are FTBS or FTFS except near sonic points where the wave speeds change sign 1 Compressive sonic points typically occur inside stationary or slowly moving shocks. 13 / 59

14 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 14 / 59 First-Order Upwind Reconstruction-Evolution Methods Euler Equations Assuming that the reconstruction is piecewise constant, each cell gives rise to a Riemann problem and the exact evolution of the piecewise constant reconstruction yields ˆF n x i+1/2 = 1 t t n+1 t n F x WRIEMANN x i+1/2, t) ) dt Again, the solution of the Riemann problem being self-similar, W RIEMANN x i+1/2, t) is constant for all time = ˆF n x i+1/2 = F x WRIEMANN x i+1/2, t) ) = F x W RIEMANN 0)) ) 4) where F x WRIEMANN x i+1/2, t) ) is computed using any exact or approximate Riemann solver and t is any arbitrary time t > t n Many first-order upwind methods for the Euler equations are based on 4) 14 / 59

15 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 15 / 59 First-Order Upwind Reconstruction-Evolution Methods Euler Equations ˆF x n i+1/2 = F x WRIEMANN x i+1/2, t) ) Again, equation 4) assumes that the waves from different cell edges do not interact or at least that any interaction does not affect the solution at the cell edges) If λρa) 1, waves travel at most one grid cell per time-step: They can interact, but the interactions cannot reach the cell edges during a single time-step Unlike in the case of scalar conservation laws where a compressive point inside the cell must be present for waves to interact when λau) 1, waves in the subsonic Euler equations interact routinely for 1/2 ρa) 1 since there are always right- and left-running waves in subsonic flows) λρa) 1 is also the CFL condition for all first-order upwind methods 15 / 59

16 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 16 / 59 First-Order Upwind Reconstruction-Evolution Methods Euler Equations Upwind methods based on ˆF n x i+1/2 = F x WRIEMANN x i+1/2, t) ) if the wave speeds are always positive as in some supersonic flows), all waves originating from x i+1/2 are right-running ˆF x n i+1/2 = F x WRIEMANN x i+1/2, t) ) = F xwi n ) FTBS if the wave speeds are always negative as in some supersonic flows), all waves originating from x i+1/2 are left-running ˆF x n i+1/2 = F x WRIEMANN x i+1/2, t) ) = F xwi+1) n FTFS the above is true for the exact Riemann solver or any reasonable approximate Riemann solver 2 all first-order upwind methods based on ˆF x n i+1/2 = F x WRIEMANN x i+1/2, t) ) are FTBS or FTFS for supersonic flows 2 Recall that F x W 0)) A RL W 0) W L ) + F x W L ) = A RL W R W L ) + F x W L ) A RL W 0) W R ) + F x W R ) = A + RL W R W L ) + F x W R ) 16 / 59

17 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 17 / 59 First-Order Upwind Reconstruction-Evolution Methods Euler Equations Most properties of first-order upwind methods for scalar conservation laws carry over to first-order upwind methods for the Euler equations The reverse is not necessarily true: for example, the Riemann problem for the Euler equations is not monotonicity preserving but that for scalar conservation laws is monotonicity preserving first-order upwind methods for the Euler equations based on reconstruction-evolution sometimes produce spurious oscillations, especially at steady and slowly moving shocks 17 / 59

18 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 18 / 59 First-Order Upwind Reconstruction-Evolution Methods Roe s First-Order Upwind Method for the Euler Equations Recall Roe s approximate Riemann solver for the Euler equations in particular, recall that the linear approximate) flux function at x = 0 can be written as A RL W 0) = 1 2 A RLW R + W L ) 1 2 replace W L and W R by W n i 3 λ k ξ k r k k=1 = 1 2 A RLW R + W L ) 1 2 A RL W R W L ) and W n i+1, respectively, and replace A RL by A n i+1/2 replace also t = 0 by t = t n, and x = 0 by x = x i+1/2 The vector version Roe s method becomes ˆF n x i+1/2 = F x WRIEMANN x i+1/2, t) ) 18 / 59

19 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 19 / 59 First-Order Upwind Reconstruction-Evolution Methods Roe s First-Order Upwind Method for the Euler Equations Let λ n i+1/2 ) j, ri+1/2 n ) j, and li+1/2 n ) j denote the eigenvalues, right eigenvectors, and left eigenvectors of A n i+1/2, respectively Let ξ n i+1/2 ) j = l n i+1/2 )T j W n i+1 W n i ) Then ˆF n x i+1/2 = 1 2 = 1 2 Fx W n i+1) + F x W n i ) ) j=1 λ n i+1/2 j ξ n i+1/2 ) jr n i+1/2 ) j Fx W n i+1) + F x W n i ) ) 1 2 An i+1/2 W n i+1 W n i ) 19 / 59

20 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 20 / 59 First-Order Upwind Reconstruction-Evolution Methods Roe s First-Order Upwind Method for the Euler Equations Recall also that the Roe-average matrix must satisfy Also recall that F x W n i+1) F x W n i ) = A n i+1/2 W n i+1 W n i ) A n+ i+1/2 + An i+1/2 = An i+1/2 and A n+ i+1/2 An i+1/2 = An i+1/2 Hence F x W n i+1) F x W n i ) = ) A n+ i+1/2 + An i+1/2 Wi+1 n Wi n ) 5) For this reason, Roe s first-order upwind method is sometimes called a flux difference splitting method Fx W n i+1) F x W n i ) ) = A n+ i+1/2 W n i+1 W n i ) + A n i+1/2 W n i+1 W n i ) 6) 20 / 59

21 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 21 / 59 Introduction to Second- & Higher-Order Reconstruction-Evolution Methods Start with the temporal evolution ˆF n x i+1/2 1 t t n+1 t n F x W xi+1/2, t) ) dt Step 1: Compute a second or even higher-order approximation of the above integral for example, use the midpoint rule 1 t n+1 F x W xi+1/2, t) ) ) dt = F x W x i+1/2, t n+1/2 ) + O t 2 ) t t n or the trapezoidal rule 1 t n+1 F x W xi+1/2, t) ) dt = 1 ) t t n 2 Fx W x i+1/2, t n+1 ) Fx W xi+1/2, t n ) ) + O t 2 ) 21 / 59

22 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 22 / 59 Introduction to Second- & Higher-Order Reconstruction-Evolution Methods Step 2: Perform a Taylor series expansion about t = t n for example F x W xi+1/2, t) ) = F x W xi+1/2, t n ) ) + F x t + O t t n ) 2) W xi+1/2, t n ) ) t t n ) Step 3: Express time derivatives in the Taylor series in terms of space derivatives for example, the time derivative of the momentum flux can be obtained from the conservation of momentum t ρv v x x) = ρv x x + v x x ρv x) p x 22 / 59

23 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 23 / 59 Introduction to Second- & Higher-Order Reconstruction-Evolution Methods Step 4: Differentiate the spatial reconstruction at time level n to approximate the spatial derivative at x i+1/2, t n ) note that in general, the reconstruction and/or its derivatives contain jump discontinuities at the cell edges at x i+1/2 average the left and right limits, W i+1/2,l t) and W i+1/2,r t), of the approximation of W x i+1/2, t) the Riemann solver average is the only average that yields the exact solution in the case of piecewise constant reconstruction use W i+1/2,l t) and W i+1/2,r t) as the left and right states in the exact solution of the Riemann problem 23 / 59

24 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 24 / 59 Introduction to Second- & Higher-Order Reconstruction-Evolution Methods Because the above four-step procedure is based on Taylor series and differential forms, it does not apply at shocks At shocks, all higher-order terms of the approximation should be eliminated and a return to first-order piecewise constant reconstruction becomes essential 24 / 59

25 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 25 / 59 The MUSCL/TVD Method MUSCL: Monotonic Upwind Scheme for Conservation Laws More specifically, this section describes the method proposed in 1986 by Anderson, Thomas, and Van Leer who called it the MUSCL method, and not the original MUSCL method designed in 1979 by Van Leer using the approach summarized in the previous section Both of this method and the original MUSCL method are reconstruction-evolution methods 25 / 59

26 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 26 / 59 The MUSCL/TVD Method The Method of Lines Two-step approximation procedure Spatial Discretization time is frozen and space is discretized F x x x i, t) ˆF xs,i+1/2 t) ˆF xs,i 1/2 t) the above equation is called the semi-discrete finite volume/difference approximation ˆF xs is called the semi-discrete conservative numerical flux the semi-discrete approximation comprises a system of ordinary differential equations in many cases, it is needed only at discrete time levels, in which case it can be written as x dw n i dt ˆF x n s,i+1/2 ˆF x n s,i 1/2 x where W n i = W x i, t n ) and ˆF n x s,i+1/2 = ˆF xs,i+1/2 t n ) 7) 26 / 59

27 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 27 / 59 The MUSCL/TVD Method The Method of Lines Two-step approximation procedure continue) dw n i dt ˆF x n s,i+1/2 ˆF x n s,i 1/2 x Temporal Discretization any ordinary differential equation solver time-integration algorithm) can be used to solve Eq. 7) in other words, space is frozen and time is discretized the resulting approximation for example using FT W n+1 i Wi n = t ˆF n x i+1/2 ˆF n x i 1/2 is called the fully discrete finite difference approximation, and ˆF x is called the fully discrete conservative numerical flux The two-stage approximation procedure described above is sometimes called the method of lines, where the lines are the coordinate lines x = cst and t = cst x 27 / 59

28 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 28 / 59 The MUSCL/TVD Method Flux Splitting In the context of the finite volume method ˆF x n s,i+1/2 = ˆF xs,i+1/2 t n ) F x W xi+1/2, t n ) ) Using flux vector splitting that is, assuming F x W ) = F x + W ) + Fx W ) Leads to ˆF x n s,i+1/2 F x + W xi+1/2, t n ) ) + F x W xi+1/2, t n ) ) Therefore, one needs to approximate F x + W xi+1/2, t n ) ) or equivalently, W x i+1/2, t n ) with a leftward bias, and W xi+1/2, t n ) ) or equivalently, W x i+1/2, t n ) with a F x rightward bias 28 / 59

29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 29 / 59 The MUSCL/TVD Method Second-Order Approximation First, W x i+1/2, t n ) is approximated with a leftward bias a first-order accurate reconstruction of the primitive variables u of which W is made leads to ux i+1/2, t n ) ū n i a second-order accurate linear reconstruction of the primitive variables u of which W is made leads to ux i+1/2, t n ) ūi n + ūn i ūi 1 n x i+1/2 x i ) = ūi n + 1 x 2 ūn i ūi 1) n instead of a pure constant or pure linear reconstruction, the following convex linear combination is considered u n+ i+1/2 = ūn i φn+ i ū n i ū n i 1) where the superscript n+ refers to time t n and the leftward bias, and the parameter φ is called a slope limiter: Its purpose is to enforce a first-order accurate reconstruction near shocks φ = 0), and a second-order accurate reconstruction φ 0) in the smooth regions of the flow 29 / 59

30 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 30 / 59 The MUSCL/TVD Method Second-Order Approximation Principle of a slope limiter 30 / 59

31 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 31 / 59 The MUSCL/TVD Method Second-Order Approximation Next, W x i+1/2, t n ) is approximated with a rightward bias a first-order accurate reconstruction of the primitive variables of which W is made leads to ux i+1/2, t n ) ū n i+1 a second-order accurate linear reconstruction of the primitive variables of which W is made leads to ux i+1/2, t n ) ūi+1+ n ūn i+2 ūi+1 n x i+1/2 x i+1 ) = ūi+1 n 1 x 2 ūn i+2 ūi+1) n instead of a pure constant or pure linear reconstruction, the following convex linear combination is considered u n i+1/2 = ūn i φn i+1ū n i+2 ū n i+1) where the superscript n refers to time t n and the rightward bias 31 / 59

32 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 32 / 59 The MUSCL/TVD Method Second-Order Approximation Applying the method of lines yields dw n i dt where ˆF x n s,i+1/2 = F x + ˆF x n s,i+1/2 ˆF x n s,i 1/2 = x ) W u n+ i+1/2 ) + Fx = F x + W ūi n W + F x ) W u n i+1/2 ) )) φn+ i ū n i ū n i 1) ū n i φn i+1 ūn i+2 ū n i+1) )) 32 / 59

33 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 33 / 59 The MUSCL/TVD Method Second-Order Approximation Many variants of the MUSCL method exist today The most popular ones use flux difference splitting along the lines of an approximate Riemann solver such as Roe s solver instead of flux vector splitting: In this case, convex linear combinations of the constant and linear reconstructions of W i+1/2,l t) and W i+1/2,r t) are used as the left and right states in the solution of the approximate Riemann problem Many slope limiters have been proposed in the literature and continue to be the subject of on-going research: One example is presented next 33 / 59

34 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 34 / 59 The MUSCL/TVD Method Van Leer s Slope Limiter Ratios of solution differences r n+ i = un i ui 1 n ui+1 n, r n un i i = un i+1 un i ui n ui 1 n = 1 r n+ i Note that r n± i 0 if the solution u is monotone increasing or monotone decreasing r n± i 0 if the solution u has a maxium or a minimum r n+ i is large and r n i is small if the solution differences decrease dramatically from left to right or if ui+1 n ui n r n+ i is small and r n i is large if the solution differences increase dramatically from left to right or if ui 1 n ui n 34 / 59

35 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 35 / 59 The MUSCL/TVD Method Van Leer s Slope Limiter Very large or very small ratios r n± i sometimes signal shocks, but not always for example, if ui+1 n ui n = 0 and ui n ui 1 n 0, then r n+ i = regardless of whether the solution is smooth or shocked in general, because there is only limited information contained in solution samples, no completely reliable way to distinguish shocks from smooth regions exists consequently, slope-limited methods do not even attempt to identify shocks: Instead, they regulate maxima and minima whether or not they are associated with shocks using the nonlinear stability conditions for example, TVD) 35 / 59

36 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 36 / 59 The MUSCL/TVD Method Van Leer s Slope Limiter Van Leer s slope limiter can be described as φr) = 2r 1 + r for r 0 0 for r < 0 Note that lim r φr) = 2 Equipping the scheme described in the previous pages with φ n± i = φr n /+ i ) explain why in this case it is φr n /+ i ) and not φr n± i )) makes it: TVD and therefore nonlinearly stable second-order accurate in monotone regions of the flow proof given in class) between first-order and second-order accurate at extrema 36 / 59

37 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 37 / 59 Steger-Warming Flux Vector Splitting Method for the Euler Equations Recall that flux splitting is defined as F x W ) = F x + W ) + Fx W ) df x + dw 0, dfx dw 0 Hence, the flux split form of the Euler equations is W t + F + x x + F x x = 0 Then, F + x x can be discretized conservatively using at least one point to the left for example, using BS and F x can be x discretized conservatively using at least one point to the right for example, using FS thus obtaining conservation and satisfaction of the CFL condition 37 / 59

38 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 38 / 59 Steger-Warming Flux Vector Splitting Method for the Euler Equations Recall the related concept of wave speed splitting AW ) = A + W ) + A W ) A + W ) 0, A W ) 0 Then, the vector conservation law can be written in wave speed split form as W + A + W W + A t x x = 0 where the matrices A + and A are usually obtained by splitting the eigenvalues of A into positive and negative parts λ i = λ + i + λ i, λ + i 0, λ i 0 = Λ = Λ + + Λ, Λ + 0, Λ 0 38 / 59

39 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 39 / 59 Steger-Warming Flux Vector Splitting Method for the Euler Equations For the 1D Euler equations QAQ 1 = Λ A = Q 1 ΛQ λ 1 = v x λ 2 = v x + c λ 3 = v x c where Q 1 = 1 v x v 2 x 2 Q = γ 1 ρc ρ 2c ρ c v 2 x ρ ρ 2c 2 + c2 γ 1 + cv x ) ρ 2c 2c v x + c) ρ 2c v x c) ) ρ v 2 x 2c 2 + c2 γ 1 cv x v 2 x 2 + c2 γ 1 ρ c v x ρ c v 2 x 2 cvx γ 1 v x + c γ 1 1 v 2 x 2 cvx γ 1 v x + c γ 1 1 A = A + + A, A + 0, A 0 A + = Q 1 Λ + Q 0, A = Q 1 Λ Q 0 ) 39 / 59

40 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 40 / 59 Steger-Warming Flux Vector Splitting Method for the Euler Equations For the Euler equations, F x is homogeneous of degree 1: Therefore F x = df x dw W = AW F ± x = A ± W = Q 1 Λ ± QW It follows that Steger and Warming, 1981) F x ± = γ 1 1 ρλ ± 1 v x γ 1 2 v x 2 + ρ 2γ λ± 2 + ρ 2γ λ± 3 1 v x + c 1 2 v x 2 + c2 γ 1 + cv x 1 v x c 1 2 v x 2 + c2 γ 1 cv x 40 / 59

41 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 41 / 59 Steger-Warming Flux Vector Splitting Method for the Euler Equations Practical implementation compute at each grid point or in each cell) λ + i = max0, λ i ) = 1 2 λ i + λ i ), λ i = min0, λ i ) = 1 2 λ i λ i ) compute at each grid point or in each cell) the Mach number M = vx c if M 1 F + x = 0, F x = F x if 1 M 0 F + x = F x = γ 1 γ ρ vx + c) 2γ ρv x 1 v x + c 1 2 v x 2 + c2 1 v x 1 2 v 2 x γ 1 + cvx + ρ vx c) 2γ 1 v x c 1 2 v 2 x + c2 γ 1 cvx 8) 41 / 59

42 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 42 / 59 Steger-Warming Flux Vector Splitting Method for the Euler Equations Practical implementation continue) continue continue if 0 M 1 F + x = γ 1 γ F x = if M > 1 ρv x 1 v x 1 2 v x 2 ρ vx c) 2γ + ρ vx + c) 2γ 1 v x c 1 2 v 2 x + c2 γ 1 cvx F + x = Fx, F x = 0 1 v x + c 1 2 v 2 x + c2 γ 1 + cvx 42 / 59

43 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 43 / 59 Steger-Warming Flux Vector Splitting Method for the Euler Equations Problem: At the sonic points M = 0 for the entropy waves and M = ±1 for the acoustic waves), the wave speed splitting 8) is not differentiable the first derivative of the split wave speeds is discontinuous at the sonic points because the function absolute value is discontinuous at zero: Consequently, the wave speed splitting 8) may experience numerical difficulties at these sonic points Solution: Regularization or entropy fix, or rounding the corner ) λ ± i = 1 ) λ i ± λ 2i + ɛ 2 2 where ɛ is a small user-adjustable parameter 43 / 59

44 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 44 / 59 Multidimensional Extensions The extension to multiple dimensions of the material covered in this chapter is straightforward except perhaps for the characteristic theory) The expressions of the Euler equations in 2D and 3D can be obtained from Chapter 2 as particular cases of the expression of the Navier-Stokes equations in 3D) Furthermore, given that Chapter 6 discusses the extension of the finite difference method to multiple dimensions on structured grids, this chapter discusses for the sake of variety the extension of the finite volume method to multiple dimensions on unstructured grids For simplicity, the focus is set here on the 2D Euler equations W t + F xw ) x + F y W ) y = 0 9) and on unstructured triangular meshes 44 / 59

45 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 45 / 59 Multidimensional Extensions 2D unstructured triangular mesh Euler-Poincaré theorem: Asymptotically, there are about two times more triangles than nodes in a 2D triangular mesh 45 / 59

46 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 46 / 59 Multidimensional Extensions 2D cell-centered grid simple implementation: each triangle is at the same time a control volume or primal cell three numerical fluxes per triangle only, but more flow variables than necessary recall the Euler-Poincaré theorem) 46 / 59

47 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 47 / 59 Multidimensional Extensions 2D vertex-based grid deemed more accurate than the cell-centered approach for stretched/skewed grids more memory efficient than comparable cell-centered techniques recall the Euler-Poincaré theorem) requires however the construction of an associated control volume or dual cell 47 / 59

48 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 48 / 59 Multidimensional Extensions 2D control volume or dual cell for the vertex-based approach Sample algorithm for constructing a dual cell C i attached to vertex i for each triangle τ ikl connected to i: determine the point of intersection of the three medians, G ikl connect the point G ikl to the midpoints M il and M ik of the edges il and ik, respectively 48 / 59

49 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 49 / 59 Multidimensional Extensions 2D control volume or dual cell for the vertex-based approach continue) at the wall boundaries, one obtains half dual cells the union of all full and half dual cells covers exactly the union of all triangles τ ikl and therefore covers exactly the given triangular mesh 49 / 59

50 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 50 / 59 Multidimensional Extensions 2D cell edge for the vertex-based approach the boundary C i of each dual cell C i attached to vertex i is the union of cell edges C i C i = C ij C ik C il C im C in the cell edge C ij, which is attached to the edge ij, is itself the union of two segments M ij G ijk and M ij G ijn 50 / 59

51 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 51 / 59 Multidimensional Extensions Consider again the 2D Euler equations 9): From the results of Chapter 3, it follows that the integration over the dual cell C i of these equations can be written after usage of the divergence theorem) as W i t + 1 C i C i F νi dγ i = 0, where W i = 1 C i C i W i dω 10) 51 / 59

52 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 52 / 59 Multidimensional Extensions Integrating Eq. 10) between t n and t n+1, dividing by t, and re-arranging gives 1 t n+1 W i t t n t dt = 1 t n+1 ) 1 F νi dγ i dt t t n C i C i = 1 t n+1 ) 1 F νi dγ i dt 11) t t n C i C i 52 / 59

53 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 53 / 59 Multidimensional Extensions Eq. 11) can be re-written as t n+1 t n W i t dt = t C i 1 t t n+1 t n C i F νi dγ i ) ) dt Which leads to W t t ) n i = t C i ˆF n i 12) where W n i 1 W i x, y, t n )dω, C i C i ˆF n i 1 t t n+1 t n C i F νi dγ i 13) Eq. 12) is the 2D extension of Eq. 1), and definitions 13) are the 2D extensions of definitions 2) ) dt 53 / 59

54 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 54 / 59 Multidimensional Extensions I { Hence, ˆF n i 1 }}{ t n+1 ) F W ) νi dγ i dt t t n C i A popular approximation of I is I F W M i, t)) ν i dγ i C i = ˆF n i 1 t n+1 ) F W Mi, t)) ν i dγ i dt is constructed in 2D similarly t t n C i to how ˆF n x 1 t n+1 F x W xi+1/2, t) ) dt in constructed in 1D i+1/2 t t n 54 / 59

55 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 55 / 59 Multidimensional Extensions In particular, upwind methods based on ˆF n i = F W RIEMANN M i, t)) ν i dγ i C i = F W RIEMANN M i, t)) ) l 1) ν 1) i + l 2) ν 2) i are the 2D extensions of 1D upwind methods based on ˆF n x i+1/2 = F x WRIEMANN x i+1/2, t) ) Let ) ν i = l 1) ν 1) i + l 2) ν 2) i 55 / 59

56 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 56 / 59 Multidimensional Extensions Computation of F W RIEMANN M ij, t)) Method #1)) straightforward extension of Roe s one-dimensional approximate Riemann solver to two or) three dimensions consider the multidimensional flux in the direction of ν ij F νij = F ν ij = ρv νij, ρv x v νij +p ν ijx, ρv y v νij +p ν ijy, ρv z v νij +p ν ijz, E+p)v νij ) T where v νij = v x νijx + v y νijy + v z νijz Roe s averages are based in this case on the secant approximation defined by F νij where A νij =, which gives W v xij = F νij W i ) F νij W j ) = A νij W ij )W i W j ) ρi v xi + ρ j v xj ρi + ρ j v yij = ρi v yi + ρ j v yj ρi v zi + ρ j v zj ρi + v zij = ρ j ρi + ρ j h ij = = c ij = H i ρi + H j ρj ρi ρi + h i + ρ j h j = ρ j ρi + ρ j γ 1) h ij 12 ) v 2xij ρ ij = ρ i ρ j the eigenvalues of A νij are: v νij ± c, and v νij with multiplicity 3 56 / 59

57 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 57 / 59 Multidimensional Extensions Computation of F W RIEMANN M ij, t)) Method #1)) straightforward extension of Roe s one-dimensional approximate Riemann solver to two or) three dimensions continue) formulate and solve Roe s multidimensional approximate Riemann problem: W + A t νij W ij ) W = 0, where s is the abscissa along ij, with uniform initial s conditions on an infinite spatial domain, except for a single jump discontinuity switch to a unitary normal ν ν ij ij = ν ij 2 compute Roe s flux ) F W RIEMANN M ij, t) = ) 1 F 2 ν W i ) + F ij ν W j ) ij 1 A 2 ν W ij ) W j W i ) ij ) ν ij 2 57 / 59

58 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 58 / 59 Multidimensional Extensions Computation of F W RIEMANN M ij, t)) Method #2)) along the edge ij, compress the conservative fluid state vector to obtain W comp = ρ, ρv n, ρe ρv 2 n )T, where v n = v ν ij, v denotes the fluid velocity vector, and comp stands for compressed along the edge ij, formulate Roe s one-dimensional approximate Riemann problem using W comp i and W comp j, and solve this problem to obtain W comp RIEMANN M ij, t) = W comp RIEMANN ρ = RIEMANN, ρv n) RIEMANN, ρe + 1 ) T 2 ρv 2 n ) RIEMANN now, given that the fluid velocity vector v can be decomposed as v = v n νij + v v n νij ), it follows that Roe s) one-dimensional }{{}}{{} normal direction tangential direction approximate) Riemann problem along the edge ij acts only on v n next, reconstruct v RIEMANN as v RIEMANN = v RIEMANNn νij + v v n νij ) = v RIEMANNx e x + v RIEMANNy e y then, compute the expanded vector W RIEMANN W exp RIEMANN = ρ RIEMANN, ρ RIEMANN v RIEMANNx, ρ RIEMANN v RIEMANNy, ρ RIEMANN e RIEMANN ρ RIEMANN v 2 RIEMANN )T where exp stands for expanded finally, compute F W RIEMANN M ij, t)) as F W exp ) RIEMANN 58 / 59

59 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 59 / 59 Multidimensional Extensions Note that ˆF ij n = ) F W RIEMANN M ij, t)) l 1) ν 1) ij + l 2) ν 2) ij ˆF ji n = ) F W RIEMANN M ij, t)) l 1) ν 1) ji + l 2) ν 2) ji Since ν ji = ν ij, it follows that ˆF ji n = ˆF ij n = ˆF n ij + ˆF n ji = 0 = ˆF ij is a conservative numerical flux 59 / 59