Computation Fluid Dynamics

Transcription

1 Computation Fluid Dynamics CFD I Jitesh Gajjar Maths Dept Manchester University Computation Fluid Dynamics p.1/189

2 Garbage In, Garbage Out We will begin with a discussion of errors. Useful to understand different types of errors which can arise when doing numerical computation. Roundoff errors Computation Fluid Dynamics p.2/189

3 Garbage In, Garbage Out We will begin with a discussion of errors. Useful to understand different types of errors which can arise when doing numerical computation. Roundoff errors Errors in modelling Computation Fluid Dynamics p.2/189

4 Garbage In, Garbage Out We will begin with a discussion of errors. Useful to understand different types of errors which can arise when doing numerical computation. Roundoff errors Errors in modelling Programming errors, ie bugs Computation Fluid Dynamics p.2/189

5 Garbage In, Garbage Out We will begin with a discussion of errors. Useful to understand different types of errors which can arise when doing numerical computation. Roundoff errors Errors in modelling Programming errors, ie bugs Truncation, discretization errors. Computation Fluid Dynamics p.2/189

6 Garbage In, Garbage Out We will begin with a discussion of errors. Useful to understand different types of errors which can arise when doing numerical computation. Roundoff errors Errors in modelling Programming errors, ie bugs Truncation, discretization errors. Computation Fluid Dynamics p.2/189

7 Measures of error Need some way to quantify errors. 2 useful measures. Computation Fluid Dynamics p.3/189

8 Measures of error Need some way to quantify errors. 2 useful measures. Absolute error If φ is an approximation to a quantity φ then the Absolute error is defined by φ φ Computation Fluid Dynamics p.4/189

9 Measures of error Need some way to quantify errors. 2 useful measures. Absolute error If φ is an approximation to a quantity φ then the Absolute error is defined by φ φ Relative error The Relative error is defined by φ φ, φ 0 φ Computation Fluid Dynamics p.4/189

10 Roundoff errors These arise when a computer is used for doing numerical computation. Computation Fluid Dynamics p.5/189

11 Roundoff errors These arise when a computer is used for doing numerical computation. Example Inexact representation of numbers eg, π, 2. Computation Fluid Dynamics p.5/189

12 Roundoff errors These arise when a computer is used for doing numerical computation. Example Inexact representation of numbers eg, π, 2. Rounding and chopping errors. Remember the way certain quantities are computed is under your control. Computation Fluid Dynamics p.5/189

13 Errors in modelling Example: Replacing full N-S equations with Euler equations. Neglect of viscous terms means no matter how accurate the numerical solution, viscous effects will not be captured where important. Computation Fluid Dynamics p.6/189

14 Programming errors, ie bugs These are all too familiar! Computation Fluid Dynamics p.7/189

15 Programming errors, ie bugs These are all too familiar! The computer is only doing what you ask it to do. Computation Fluid Dynamics p.7/189

16 Programming errors, ie bugs These are all too familiar! The computer is only doing what you ask it to do. Even NASA has made blunders. Computation Fluid Dynamics p.7/189

17 Subtle errors Suppose φ = O(10 8 ), φ = O(10 8 ) then something like diff = MAX(ABS(phi-phistar)) tol= 1.e-6 IF(diff < tol) EXIT in numerical codes will be wrong usage. The condition is always satisfied even though relative error is O(1). Computation Fluid Dynamics p.8/189

18 Truncation, discretization errors. A pproximate by U xx = f(x) (u(x i+1 ) 2u(x i ) + u(x i 1 )) h 2 = f(x i ). Computation Fluid Dynamics p.9/189

19 Truncation, discretization errors. A pproximate by U xx = f(x) (u(x i+1 ) 2u(x i ) + u(x i 1 )) h 2 = f(x i ). This gives rise to a truncation error ie Computation Fluid Dynamics p.9/189

20 Truncation, discretization errors. A pproximate by U xx = f(x) (u(x i+1 ) 2u(x i ) + u(x i 1 )) h 2 = f(x i ). This gives rise to a truncation error ie τ(x i ) = U xx (x i ) f(x i ) = h2 12 U xxxx (x i ) +... Computation Fluid Dynamics p.9/189

21 Initial value problems Here we will look at the solution of ordinary differential equations of the type, say dy dx = f(x, y), a x b, subject to an initial condition y(a) = α Computation Fluid Dynamics p.10/189

22 Example Solve dy dx = y(1 y ), 0 x, 4 subject to an initial condition y(0) = 1 Computation Fluid Dynamics p.11/189

23 Soln of ODE s The methods also generalise to systems of equations i.e. dy = F(x, Y), dx a x b, where Y = (y 1 (x), y 2 (x),..., y N (x)) T, F = (f 1 (x, Y), f 2 (x, Y),..., f N (x, Y)) T, with initial data Y(a) = α, say, where α = (α 1, α 2,..., α N ) T. Computation Fluid Dynamics p.12/189

24 Example Solve y 2xyy + y 2 = 1, y(1) = 1, y (1) = 2. The equivalent first order system is obtained with (y 1 (x), y 2 (x)) T = (y(x), y (x)) T, f 1 (x, y 1, y 2 ) = y 2 (x), f 2 (x, y 1, y 2 ) = 1 + 2xy 1 (x)y 2 (x) y1(x), 2 and initial condition (y 1 (1), y 2 (1)) T = (1, 2) T. Computation Fluid Dynamics p.13/189

25 A mathematical result. Suppose we define D to be the domain D = {(x, y) a x b, < y < } and f(x, y) is continuous on D. If f(x, y) satisfies a Lipschitz condition on D then the ODE has a unique solution for a x b. Computation Fluid Dynamics p.14/189

26 A mathematical result. Suppose we define D to be the domain D = {(x, y) a x b, < y < } and f(x, y) is continuous on D. If f(x, y) satisfies a Lipschitz condition on D then the ODE has a unique solution for a x b. Recall f(x, y) satisfies a Lipschitz condition on D means that there exists a constant L > 0 (called the Lipschitz constant) such that f(x 1, y 1 ) f(x 2, y 2 ) L y 1 y 2 whenever (x 1, y 1 ), (x 2, y 2 ) belong to D. Computation Fluid Dynamics p.14/189

27 Euler s Method This is the simplest of techniques for the numerical solution of ODE s. For simplicity define an equally spaced mesh x j = a + jh, j = 0,.., N where h = (b a)/n is called the step size. h a = x 0 x 1 x i x i+1 x N = b We can derive Euler s method as follows. Computation Fluid Dynamics p.15/189

28 Euler s Method Suppose y(x) is the unique solution to the ODE, and twice differentiable. Then by Taylor s theorem we have y(x i+1 ) = y(x i + h) = y(x i ) + y (x i )h + h2 2 y (ξ) where x i ξ x i+1. Computation Fluid Dynamics p.16/189

29 Euler s Method But from the differential equation y (x i ) = f(x i ), and y i = y(x i ). This suggests the scheme for calculating the w i. This is Euler s method w 0 = α w i+1 = w i + hf(x i, w i ), i = 1, 2,.., N 1, Computation Fluid Dynamics p.17/189

30 Truncation error for Euler s method Suppose that y i = y(x i ) is the exact solution at x = x i. Then the truncation error is defined by τ i+1 (h) = y i+1 (y i + hf(x i, y i )) h for i = 0, 1,..., N 1. = y i+1 y i h f(x i, y i ), Computation Fluid Dynamics p.18/189

31 Truncation error From the above we find that τ i+1 (h) = h 2 y (ξ i ) for some ξ i in (x i, x i+1 ). So if y (x) is bounded by a constant M in (a, b) then τ i+1 (h) h 2 M. Thus we see that the truncation error for Euler s method is O(h). Computation Fluid Dynamics p.19/189

32 In general if τ i+1 = h p we say that the method is of order h p. In principle if h decreases, we should be able to achieve greater accuracy, although in practice roundoff error limits the smallest size of h that we can take. Computation Fluid Dynamics p.20/189

33 Higher order methods, Modified Euler The modified Euler method is given by w 0 = α k 1 = hf(x i, w i ), w i+1 = w i + h 2 [f(x i, w i ) + f(x i+1, w i+1 )], i = 1, 2,.., N 1, This has truncation error O(h 2 ). Sometimes this is also called a Runge-Kutta method of order 2. Computation Fluid Dynamics p.21/189

34 Modified Euler Notice Euler s method is implicit w 0 = α k 1 = hf(x i, w i ), w i+1 = w i + h 2 [f(x i, w i ) + f(x i+1, w i+1 )], i = 1, 2,.., N 1, Computation Fluid Dynamics p.22/189

35 Modified Euler Notice Euler s method is implicit w 0 = α k 1 = hf(x i, w i ), w i+1 = w i + h 2 [f(x i, w i ) + f(x i+1, w i+1 )], i = 1, 2,.., N 1, Thus some iteration may be necessary. Computation Fluid Dynamics p.22/189

36 Runge-Kutta method of order 4 One of the most common Runge-Kutta methods of order 4 is given by w 0 = α, k 1 = hf(x i, w i ), k 2 = hf(x i + h 2, w i k 1) k 3 = hf(x i + h 2, w i k 2) k 4 = hf(x i+1, w i + k 3 ) w i+1 = w i (k 1 + 2k 2 + 2k 3 + k 4 ), for i = 0, 1,.., N 1 Computation Fluid Dynamics p.23/189

37 Systems of equations All these methods generalise to a system of first order equations. Thus for instance the RK(4) method above becomes Computation Fluid Dynamics p.24/189

38 Runge-Kutta 4th order w 0 = α, k 1 = hf(x i, w i ), k 2 = hf(x i + h 2, w i k 1) k 3 = hf(x i + h 2, w i k 2) k 4 = hf(x i+1, w i + k 3 ) w i+1 = w i (k 1 + 2k 2 + 2k 3 + k 4 ), for i = 0, 1,.., N 1 Computation Fluid Dynamics p.25/189

39 m-step multi-step method The methods discussed above are called one-step methods. Methods that use the approximate values at more than one previous mesh point are called multi-step methods. There are two distinct types worth mentioning. These are of the form Computation Fluid Dynamics p.26/189

40 Two types w i+1 = c m 1 w i + c m 2 w i c 0 w i+1 m +h[b m f(x i+1, w i+1 ) + b m 1 f(x i, w i ) b 0 f(x i+1 m, w i+1 m )] If b m = 0 so that there is no term with w i+1 on the right hand side of (*), the method is explicit. If b m 0 we have an implicit method. Computation Fluid Dynamics p.27/189

41 Adams-Bashforth 4th order method (explicit) Here we have w 0 = α, w 1 = α 1, w 2 = α 2, w 3 = α 3, where these values are obtained using other methods such as RK(4) for instance. Then for i = 3, 4,..., N 1 we use w i+1 = w i + h 24 [55f(x i, w i ) 59f(x i 1, w i 1 ) + 37f(x i 2, w i 2 ) 9f(x i 3, w i 3 )]. Computation Fluid Dynamics p.28/189

42 Boundary Value Problems - Shooting Methods Consider the differential equation d 2 y dx 2 + k dy dx + xy = 0, y(0) = 0, y(1) = 1. This is an example of a boundary value problem. Why? Conditions have to satisfied at both ends. Computation Fluid Dynamics p.29/189

43 BVP If we write this as a system of first order equations we have Y 1 = y, Y 2 = dy dx, dy 1 dx = Y 2, dy 2 dx = ky 2 xy 1. The boundary conditions give Y 1 (0) = 0, Y 1 (1) = 1. We do not know the value of Y 2 (0). Computation Fluid Dynamics p.30/189

44 BVP Suppose we guess the value of Y 2 (0) = g, say. Then we can integrate the system with the initial condition Y(0) = ( Y1 (0) Y 2 (0) ) = ( 0 g ), Computation Fluid Dynamics p.31/189

45 BVP This will give us Y(1) = Y 1(1) Y 2 (1) = β 1 β 2, Computation Fluid Dynamics p.32/189

46 BVP This will give us Y(1) = Y 1(1) Y 2 (1) = β 1 β 2, But β 1 will not necessarily satisfy the required condition Y 1 (1) = 1, So now need to iterative to try and get the correct value of g such that the required condition at x = 1 is satisfied. Computation Fluid Dynamics p.32/189

47 BVP, shooting To do this define φ(g) = Y 1 (1; g) 1, We want to find the value of g such that φ(g) = 0. This gives rise to the idea of a shooting method. Computation Fluid Dynamics p.33/189

48 Shooting Method Suppose that we have a guess g and we seek a correction dg such that φ( g + dg) = 0. By Taylor expansion we have φ( g + dg) = φ( g) + dφ dg ( g)dg + O(dg2 ). This suggests that we take dg = φ( g) φ ( g), and hence a new value for g is g + dg. Computation Fluid Dynamics p.34/189

49 Shooting- secant method Hence g n+1 = g n φ(g n) φ (g n ), Computation Fluid Dynamics p.35/189

50 Secant method Now we are required to find φ (g n ). How can we do this? One way is to estimate φ (g n ) by This gives φ (g n ) = φ(g n) φ(g n 1 ) g n g n 1. g n+1 = g n φ(g n)(g n g n 1 ) φ(g n ) φ(g n 1 ), which is known as the secant method. Computation Fluid Dynamics p.36/189

51 Shooting- Newton s method Consider again with Y(0) = (0, g) T. Now dy 1 dx = Y 2, dy 2 dx = ky 2 xy 1, φ (g n ) = Y 1 g (1; g), Thus differentiate the original system of equations and boundary conditions with respect to g. Computation Fluid Dynamics p.37/189

52 Shooting- Newton s method d dx ( Y g d dx ) ( Y g Y 2 g = k Y 2 x Y 1 g g ) x=0 = 0. 1, The system defines another initial value problem with given initial conditions. Computation Fluid Dynamics p.38/189

53 Shooting- Newton s method Note that φ (g) = Y 1 g (1; g). From the solution of the above we can extract (x = 1) and hence compute dg update g. Y 1 g This forms the basis of Newton s method combined with shooting, to solve boundary value problems. Computation Fluid Dynamics p.39/189

54 Newton- augmented system Can also use an augmented system where we define Y 1 = y, and then d dx Y 1 Y 2 Y 3 Y 4 Y 2 = dy dx, = Y 2 ky 2 xy 1 Y 4 ky 4 xy 3 Y 3 = y g = Y 1 g, Y 4 = Y 2 g,, Y(0) = Y 1 (0) Y 2 (0) Y 3 (0) Y 4 (0) = 0 g 0 1 Computation Fluid Dynamics p.40/189

55 Multiple shooting Consider d 4 y dx = 4 y3 ( dy dx )2, dy y(0) = 1, dx (0) = 0, y(1) = 2, dy (1) = 1, dx We need two starting values at x = 0 and then we will have two conditions to satisfy at x = 1. Computation Fluid Dynamics p.41/189

56 Multiple shooting Define Y 1 = y, Y 2 = y, Y 3 = y, y 4 = y, We will need to guess for Y 3 (0) = e, say, and Y 4 (0) = g. Computation Fluid Dynamics p.42/189

57 Multiple shooting φ 1 (e, g) = Y 1 (x = 1; e; g) 2, φ 2 (e, g) = Y 2 (x = 1; e; g) 1. We need to iterative on both c and g to ensure that the remaining conditions are satisfied. To find corrections, need Taylor expansion for function of two variables. Computation Fluid Dynamics p.43/189

58 Multiple shooting To obtain the corrections to guessed values ẽ, g we have φ 1 (ẽ + de, g + dg) = 0 = φ 1 (ẽ, g) + de φ 1 e (ẽ, g) + dg φ 1 g (ẽ, φ 2 (ẽ + de, g + dg) = 0 = φ 2 (ẽ, g) + de φ 2 e (ẽ, g) + dg φ 2 g (ẽ, Computation Fluid Dynamics p.44/189

59 Multiple shooting Multidimensional case Vector of guesses g. We can find the corrections dg as dg = J 1 ( g)φ( g), and φ is the vector of con- where J is the Jacobian φ i g k ditions. Computation Fluid Dynamics p.45/189

60 Richardson Extrapolation Suppose that we use a method with truncation error of O(h m ) to compute an approximation w i. We can use Richardson extrapolation to get an approximation with greater accuracy. Computation Fluid Dynamics p.46/189

61 Richardson extrapolation Suppose w (1) i is approximation with step size h and w (2) i with step size 2h. Then we can write and w (1) i = y i + Eh m + E 1 h m , w (2) i = y i + E(2h) m + E 1 (2h) m Computation Fluid Dynamics p.47/189

62 Richardson extrapolation Then we can eliminate the E term to get Thus 2 m w (1) i w (2) i = (2 m 1)y i + O(h m+1 ). wi = 2m w (1) i w (2) i 2 m 1 is a more accurate approximationto the solution than w (1) i or w (2) i. Computation Fluid Dynamics p.48/189

63 Richardson extrapolation Then we can eliminate the E term to get Thus 2 m w (1) i w (2) i = (2 m 1)y i + O(h m+1 ). wi = 2m w (1) i w (2) i 2 m 1 is a more accurate approximationto the solution than w (1) i or w (2) i. For a 4th order Runge-Kutta method the above gives wi = 16w(1) i w (2) i. 15 Computation Fluid Dynamics p.48/189

64 Solution of BVP using finitedifferences Boundary value problems can also be tackled directly using finite-differences or some other technique such as spectral approximation. We will look at one specific example with finite-differences. Consider d 2 y dx = (32 + 2x3 y dy ), 1 x 3, dx y(1) = 17, y(3) = The exact solution is y(x) = x 2 + (16/x). Computation Fluid Dynamics p.49/189

65 Solution of BVP using finitedifferences Define a uniform grid (x 0, x 1,..., x N ) with N + 1 points. Grid spacing h = (x N x 0 )/N, x j = x 0 + jh, for (j = 0, 1,.., N). Computation Fluid Dynamics p.50/189

66 Solution of BVP using finitedifferences Approximate y at each of the nodes x = x i by w i The derivatives of y in the ode are approximated in finite-difference form as ( ) dy = w i+1 w i 1 + O(h 2 ), dx x=x i 2h ( d 2 ) y dx 2 x=x i = w i+1 2w i + w i 1 h 2 + O(h 2 ). Computation Fluid Dynamics p.51/189

67 Solution of BVP using finitedifferences These can be derived by making use of a Taylor expansion about the point x = x i. Thus for example y(x i+1 ) = y(x i ) + h dy dx (x i) + h2 2 y(x i 1 ) = y(x i ) h dy dx (x i) + h2 2 d 2 y dx (x i) + h3 2 6 d 2 y dx (x i) h3 2 6 d 3 y dx 3 (x i) + h4 24 d 3 y dx 3 (x i) + h4 24 d d d d By adding and subtracting and replacing y(x i ) by w i we obtain previous approximation. Computation Fluid Dynamics p.52/189

68 Solution of BVP using finitedifferences Next replace y and its derivatives in ode by the above approximations to get ( ) w i+1 2w i + w i 1 = 4 + x3 i h 2 4 w wi+1 w i 1 i, 16h for (i = 1, 2,...N 1) and w 0 = 17, w N = The above equations are a set of nonlinear difference equations. We have N + 1 equations for N + 1 unknowns w 0,..., w N. Computation Fluid Dynamics p.53/189

69 Solution of BVP using finitedifferences The nonlinear term above can now be tackled in many different ways. Thus for example we can replace it by ( (k) w w (k 1) i+1 ) w(k) i 1 i, 16h or w (k) i ( ) (k 1) w i+1 w (k 1) i 1 16h, Computation Fluid Dynamics p.54/189

70 Newton linearization Suppose that we have a guess for the solutions w i = W i. We seek corrections δw i such that the w i = W i + δw i satisfies the system. = W i + δw i into equations and lin- Substituting w i earizing gives Computation Fluid Dynamics p.55/189

71 BVP FD methods and δw i+1 2δw i + δw i 1 = F h 2 i ( ) ( ) Wi+1 W i 1 δwi+1 δw i 1 δw i W i, 16h 16h for (i = 1, 2,...N 1) δw 0 = F 0, δw N = F N. Computation Fluid Dynamics p.56/189

72 BVP, FD methods The techniques described above lead to the solution of a tridiagonal systems of linear equations of the form α i w i 1 + β i w i + γ i w i+1 = δ i, i = 0, 1,.., N, where the α i, β i, γ i are coefficients obtainable from the difference equation. Computation Fluid Dynamics p.57/189

73 BVP, FD methods For example, we have α i = 1 h 2 W i 16h, β i = 2 h 2 + W i+1 W i 1 16h, γ i = 1 h 2 + W i 16h, δ i = F i, i = 1, 2,.., N 1, and β 0 = 1, γ 0 = 0, δ 0 = F 0, α N = 0, β N = 1, δ N = F N. Computation Fluid Dynamics p.58/189

74 Thomas s tridiagonal algorithm This version of a tridiagonal solver is based on Gaussian elimination. First we create zeros below the diagonal and then once we have a triangular matrix, we solve for the w i using back substitution. Thus the algorithm takes the form Computation Fluid Dynamics p.59/189

75 Thomas s tridiagonal algorithm β j = β j γ j 1α j β j 1 j = 1, 2, 3,..., N, δ j = δ j δ j 1α j β j 1, j = 1, 2, 3,..., N, w N = δ N β N, w j = (δ j γ j w j+1 ) β j, j = N 1,..., 1, 0. Computation Fluid Dynamics p.60/189

76 Stability In practice most initial value integrators should work reasonably well on standard problems. However certain types of problems (stiff problems) can cause difficulty and care needs to be exercised in the choice of the method. Computation Fluid Dynamics p.61/189

77 Stability -Consistency A method is said to be consistent if the local truncation error tends to zero as the step size 0, i.e lim max h 0 i τ i (h) = 0. Computation Fluid Dynamics p.62/189

78 Stability -Convergence A method is said to be convergent with respect to the equation it approximates if lim max h 0 i w i y(x i ) = 0, where y(x) is the exact solution and w i an approximation produced by the method. Computation Fluid Dynamics p.63/189

79 Stability, Theorem It can be proven that if the difference method is consistent with the differential equation, then the method is stable if and only if the method is convergent. Computation Fluid Dynamics p.64/189

80 Stability of m-step methods If we consider an m-step method w 0 = α 0, w 1 = α 1,..., w m 1 = α m 1, w i+1 = a m 1 w i + a m 2 w i a 0 w i+1 m +h[f (x i, w i+1, w i,..., w i+1 m )] Computation Fluid Dynamics p.65/189

81 Stability of m-step methods If we consider an m-step method w 0 = α 0, w 1 = α 1,..., w m 1 = α m 1, w i+1 = a m 1 w i + a m 2 w i a 0 w i+1 m +h[ F (x i, w i+1, w i,..., w i+1 m )] Then ignoring the F term the homogenous part is just a difference equation. Computation Fluid Dynamics p.65/189

82 Stability of m-step method The stability is thus connected with the the roots of the characteristic polynomial Why? λ m a m 1 λ m 1... a 1 λ a 0 = 0. Computation Fluid Dynamics p.66/189

83 Stability of m-step method Consider the ODE with f(x, y) = 0. dy dx = f(x, y), a x b, y(a) = α This has the solution y(x) = α. The difference equation has to produce the same solution, ie w n = α. Computation Fluid Dynamics p.67/189

84 Stability of m-step method Next consider w i+1 = a m 1 w i + a m 2 w i a 0 w i+1 m. Computation Fluid Dynamics p.68/189

85 Stability of m-step method Next consider w i+1 = a m 1 w i + a m 2 w i a 0 w i+1 m. If we look for solutions of the form w n = λ n then this gives the characteristic polynomial equation Computation Fluid Dynamics p.68/189

86 Stability of m-step method Next consider w i+1 = a m 1 w i + a m 2 w i a 0 w i+1 m. If we look for solutions of the form w n = λ n then this gives the characteristic polynomial equation λ m a m 1 λ m 1... a 1 λ a 0 = 0. Computation Fluid Dynamics p.68/189

87 Stability of m-step method Suppose λ 1,...λ m are distinct roots of characteristic polynomial. Then we can write Computation Fluid Dynamics p.69/189

88 Stability of m-step method Suppose λ 1,...λ m are distinct roots of characteristic polynomial. Then we can write m w n = c i λ m i. i=1 Computation Fluid Dynamics p.69/189

89 Stability of m-step method Suppose λ 1,...λ m are distinct roots of characteristic polynomial. Then we can write m w n = c i λ m i. i=1 Since w n = α is a solution, the difference equation gives α a m 1 α... a 0 α = 0, or α(1 a m 1... a 0 ) = 0. Computation Fluid Dynamics p.69/189

90 Stability of m-step method Thus w n = α + m c i λ n i. i=2 Computation Fluid Dynamics p.70/189

91 Stability of m-step method Thus w n = α + m i=2 c i λ n i. In the absence of round-off error all the c i would be zero. Computation Fluid Dynamics p.70/189

92 Stability of m-step method Thus w n = α + m i=2 c i λ n i. In the absence of round-off error all the c i would be zero. If λ i 1 then the error due to roundoff will not grow. Hence the method is stable if λ i 1. Computation Fluid Dynamics p.70/189

93 Stability of m-step method Is it enough just to have stability as defined above? Computation Fluid Dynamics p.71/189

94 Stability of m-step method Is it enough just to have stability as defined above? Consider the solution of dy dx = 30y, y(0) = 1/3. Computation Fluid Dynamics p.71/189

95 Satbility of m-step method The RK(4) method, although stable, has difficulty in computing the accurate solution of this problem. This means that we need something more than just the idea of stability defined above. Computation Fluid Dynamics p.72/189

96 Absolute stability Consider dy dx = ky, y(0) = α, k < 0. The exact solution of this is y(x) = αe kx. Computation Fluid Dynamics p.73/189

97 Absolute stability Consider dy dx = ky, y(0) = α, k < 0. The exact solution of this is y(x) = αe kx. If we take our one-step method and apply it to this equation we obtain w i+1 = Q(hk)w i. Computation Fluid Dynamics p.73/189

98 Absolute Stability Similarly a multi-step of the type used earlier, when applied to the test equation gives w i+1 = a m 1 w i + a m 2 w i a 0 w i+1 m +h[b m kw i+1 + b m 1 kw i b 0 kw i+1 m ]. Computation Fluid Dynamics p.74/189

99 Absolute Stability Thus if we seek solutions of the form w i = z i this will give rise to the characteristic polynomial equation where Q(z, hk) = 0, Computation Fluid Dynamics p.75/189

100 Absolute Stability Thus if we seek solutions of the form w i = z i this will give rise to the characteristic polynomial equation where Q(z, hk) = 0, Q(z, hk) = (1 hkb m )z m (a m 1 + hkb m 1 )z m 1... (a 0 + hkb 0 ). Computation Fluid Dynamics p.75/189

101 Absolute Stability The region R of absolute stability for a one-step method is defined as the region in the complex plane R= {hk C, Q(hk) < 1}. Computation Fluid Dynamics p.76/189

102 Absolute Stability The region R of absolute stability for a one-step method is defined as the region in the complex plane R= {hk C, Q(hk) < 1}. For a multi-step method R = {hk C, β j < 1}, where β j is a root of Q(z, hk) = 0. Computation Fluid Dynamics p.76/189

103 Absolute Stability A numerical method is A-stable if R contains the entire left half plane. Computation Fluid Dynamics p.77/189

104 Consider the modified Euler method w 0 = α k 1 = hf(x i, w i ), w i+1 = w i + h 2 [f(x i, w i ) + f(x i+1, w i+1 )], This is an A-stable method. Computation Fluid Dynamics p.78/189

105 Numerical Solution of PDes Computation Fluid Dynamics p.79/189

106 Classification of PDE s Partial differential equations can be classified as being of type elliptic, parabolic or hyperbolic. In some cases equations can be of mixed type. Consider A 2 φ x 2 + B 2 φ x y + C 2 φ y 2 + D φ x + E φ y + F φ + G = 0, where, in general, A, B, C, D, E, F, and G are functions of the independent variables x and y and of the dependent variables φ. Computation Fluid Dynamics p.80/189

107 PDE s Classification The equation is said to be elliptic if B 2 4AC < 0, Computation Fluid Dynamics p.81/189

108 PDE s Classification The equation is said to be elliptic if B 2 4AC < 0, parabolic if B 2 4AC = 0, or Computation Fluid Dynamics p.81/189

109 PDE s Classification The equation is said to be elliptic if B 2 4AC < 0, parabolic if B 2 4AC = 0, or hyperbolic if B 2 4AC > 0. Computation Fluid Dynamics p.81/189

110 PDE s Classification The equation is said to be elliptic if B 2 4AC < 0, parabolic if B 2 4AC = 0, or hyperbolic if B 2 4AC > 0. Computation Fluid Dynamics p.81/189

111 Classification An example of an elliptic equation is Poisson s equation 2 φ x + 2 φ = f(x, y). 2 y2 Computation Fluid Dynamics p.82/189

112 Classification The heat equation φ t = φ k 2 x 2 is of parabolic type, and the wave equation is a hyperbolic pde. 2 φ x 2 2 φ y 2 = 0 Computation Fluid Dynamics p.83/189

113 Classification An example of a mixed type equation is the transonic small disturbance equation given by (K φ x ) 2 φ x φ y 2 = 0. Computation Fluid Dynamics p.84/189

114 Classification 2 Consider a system of first order partial differential equations. Unknowns U = (u 1, u 2,..., u n ) T Independent variables x = (x 1, x 2,..., x m ) T. Computation Fluid Dynamics p.85/189

115 Classification 2 Suppose that the equations can be written in quasi-linear form m k=1 A k U x k = Q where the A k are (n n) matrices and Q is an (n 1) column vector, and both can depend on x k and U but not on the derivatives of U. Computation Fluid Dynamics p.86/189

116 Classification 2 If we seek plane wave solutions of the homogeneous part of the above pde in the form U = U o e ix.s, where s = (s 1, s 2,.., s m ) T, then i [ m k=1 A k s k ] U = 0. Computation Fluid Dynamics p.87/189

117 Classification 2 This will have a non-trivial solution only if the characteristic equation det m k=1 A k s k = 0, Computation Fluid Dynamics p.88/189

118 Classification 2 The system is hyperbolic if n real characteristics exist. Computation Fluid Dynamics p.89/189

119 Classification 2 The system is hyperbolic if n real characteristics exist. If all the characteristics are complex, the system is elliptic. Computation Fluid Dynamics p.89/189

120 Classification 2 The system is hyperbolic if n real characteristics exist. If all the characteristics are complex, the system is elliptic. If some are real and some complex, the system is of mixed type. Computation Fluid Dynamics p.89/189

121 Classification 2 The system is hyperbolic if n real characteristics exist. If all the characteristics are complex, the system is elliptic. If some are real and some complex, the system is of mixed type. If the system is of rank less than n, then we have a parabolic system. Computation Fluid Dynamics p.89/189

122 Classification 2 The system is hyperbolic if n real characteristics exist. If all the characteristics are complex, the system is elliptic. If some are real and some complex, the system is of mixed type. If the system is of rank less than n, then we have a parabolic system. Computation Fluid Dynamics p.89/189

123 Consistency, convergence and Lax equivalence theorem Consistent A discrete approximation to a partial differential equation is said to be consistent if in the limit of the stepsize(s) going to zero, the original pde system is recovered, ie the truncation error approaches zero. Computation Fluid Dynamics p.90/189

124 Consistency, convergence and Lax equivalence theorem Stability If we define the error to be the difference between the computed solutions and the exact solution of the discrete approximation, then the scheme is stable if the error remains uniformly bounded for successive iterations. Computation Fluid Dynamics p.91/189

125 Consistency, convergence and Lax equivalence theorem Convergence A scheme is stable if the solution of the discrete equations approaches the solution of the pde in the limit that the step-sizes approach zero. Lax s Equivalence Theorem For a well posed initial-value problem and a consistent discretization, stability is the necessary and sufficient condition for convergence. Computation Fluid Dynamics p.92/189

126 Difference formulae Suppose that we have a grid of points with equal mesh spacing x in the x direction and equal spacing y in the y direction. Thus we can define points x i, y j by x i = x 0 + i x, y j = y 0 + j y. Computation Fluid Dynamics p.93/189

127 Difference formulae Suppose that we are trying to approximate a derivative of a function φ(x, y) at the points x i, y j. Denote the approximate value of φ(x, y) at the point x i, y j by w i,j say. Computation Fluid Dynamics p.94/189

128 Central Differences The first and second derivatives in x or y may be approximated as before by ( 2 ) φ = w i+1,j 2w i,j + w i 1,j + O(( x 2 ( x ) 2 x ) 2 ), ij ( 2 ) φ = w i,j+1 2w i,j + w i,j 1 + O(( y 2 ij ( y ) 2 y ) 2 ), ( ) φ = w i+1,j w i 1,j + O(( x ) 2 ), x ij 2 x ( ) φ = w i,j+1 w i,j 1 + O(( y ) 2 ). y 2 y ij Computation Fluid Dynamics p.95/189

129 Central Differences The approximations listed above are centered at the points (x i, y j ), and are called central-difference approximations. i 1, j i, j i + 1, j Computation Fluid Dynamics p.96/189

130 One-sided approximations We can also construct one-sided approximations to derivatives. Thus for example a second-order forward approximation to φ x at the point (x i, y j ) is given by ( ) φ x ij = 3w i,j + 4w i+1,j w i+2,j 2 x. Computation Fluid Dynamics p.97/189

131 Weights for central differences Node Points Order of i 2 i 1 i i + 1 i + 2 Accuracy 1st derivative ( x ) ( x ) nd derivative ( x ) ( x ) Computation Fluid Dynamics p.98/189

132 Weights for one-sided differences Computation Fluid Dynamics p.99/ Node Points Order of i i + 1 i + 2 i + 3 i + 4 Accuracy 1st derivative ( x ) -1 1 ( x ) ( x ) ( x ) nd derivative ( x ) ( x )

133 Mixed derivatives For finding suitable discrete approximations for mixed derivatives use a multidimensional Taylor expansion. Thus for example second order approximations to 2 φ/ x y at the point i, j are given 2 φ x y = w i+1,j+1 w i 1,j+1 + w i 1,j 1 w i+1,j 1 4 x y +O(( x ) 2, ( In stencil form we can express this as x y Computation Fluid Dynamics p.100/189

134 Mixed derivatives Alternatively, 2 φ x y = w i+1,j+1 w i+1,j w i,j+1 + w i 1,j 1 w i 1,j w i,j 1 + 2w i,j 2 x y + or 1 2 x y Computation Fluid Dynamics p.101/189

135 Central, one-sided differences Consider the approximation ( ) φ = w i+1,j w i,j. x x ij By Taylor expansion we see that this gives rise to a truncation error of O( x ). In addition this approximation is centered at the point x i+ 1 2,j. i, j i + 1, j Computation Fluid Dynamics p.102/189

136 Solution of elliptic pde s A prototype elliptic pde is Poisson s equation given by 2 φ x + 2 φ = f(x, y), 2 y2 where f(x, y) is a known/given function. The equation has to be solved in a domain D Computation Fluid Dynamics p.103/189

137 Boundary Conditions Boundary conditions are given on the boundary δd of D. Computation Fluid Dynamics p.104/189

138 Boundary Conditions These can be of three types: Dirichlet φ = g(x, y) on δd. Computation Fluid Dynamics p.105/189

139 Boundary Conditions These can be of three types: Dirichlet Neumann φ = g(x, y) on δd. φ n = g(x, y) on δd. Computation Fluid Dynamics p.105/189

140 Boundary Conditions These can be of three types: Dirichlet Neumann φ = g(x, y) on δd. φ n = g(x, y) on δd. Robin/Mixed B(φ, φ ) = 0 on δd. n Robin boundary conditions involve a linear combination of φ and its normal derivative on the boundary. Mixed boundary conditions involve different conditions for one part of the boundary, and another type for other parts of the boundary. Computation Fluid Dynamics p.105/189

141 Solution of model problem Let us consider a model problem with 2 φ x + 2 φ = f(x, y), 0 < x, y < 1 2 y2 φ = 0 on δd. Here the domain D is the square region 0 < x < 1 and 0 < y < 1. Computation Fluid Dynamics p.106/189

142 Solution of model problem Construct a finite difference mesh with points (x i, y j ), say where x i = i x, i = 0, 1,..., N, y j = j y, j = 0, 1,... where x = 1/N, and y = 1/M are the step sizes in the x and y directions. Computation Fluid Dynamics p.107/189

143 Solution of model problem Next replace the derivatives in Poisson equation by the discrete approximations to get w i+1,j 2w i,j + w i 1,j ( x ) 2 + w i,j+1 2w i,j + w i,j 1 ( y ) 2 = f i,j, and 1 i N 1, 1 j M 1 w i,j = 0, if i = 1, N, 1 j M, w i,j = 0, if j = 1, M, 1 i N. Computation Fluid Dynamics p.108/189

144 Solution of model problem Thus we have (N 1) (M 1) unknown values w i,j to find at the interior points of the domain. Computation Fluid Dynamics p.109/189

145 If we write w i = (w i,1, w i,2,..., w i,m 1 ) T and f i = (f i,1, f i,2,..., f i,m 1 ) T we can write the above system of equations in matrix form as Computation Fluid Dynamics p.110/189

146 Solution of model problem B I I B I I B I I B w 1 w 2 w 3 w N 1 = ( x ) 2 f 1 f 2 f 3 f N 1 Computation Fluid Dynamics p.111/189

147 Solution of model problem In the above I is the (M 1) (M 1) identity matrix and with b = 2( B = b c c b c 0 c b c... c b 3, ( x ) ( y ) 2), c = 1. Computation Fluid Dynamics p.112/189

148 Solution of model problem Let us write the linear system as Aw = f What we observe is that the matrix A is very sparse. The matrix A is very large. For instance with N = M = 101 the linear system is of size and we have 10 4 unknowns to find. Computation Fluid Dynamics p.113/189

149 Solution of linear system The linear system can be solved using Direct Methods These are not efficient if A is large. Also very expensive, and require large storage. Computation Fluid Dynamics p.114/189

150 Solution of linear system The linear system can be solved using Direct Methods These are not efficient if A is large. Also very expensive, and require large storage. Iterative Methods The method of choice for most applications. Computation Fluid Dynamics p.114/189

151 Iterative methods Point relaxation. Jacobi, Gauss-Seidel, SOR Computation Fluid Dynamics p.115/189

152 Iterative methods Point relaxation. Jacobi, Gauss-Seidel, SOR Line Relaxation Computation Fluid Dynamics p.115/189

153 Iterative methods Point relaxation. Jacobi, Gauss-Seidel, SOR Line Relaxation Conjugate gradient and PCG methods Computation Fluid Dynamics p.115/189

154 Iterative methods Point relaxation. Jacobi, Gauss-Seidel, SOR Line Relaxation Conjugate gradient and PCG methods Minimal residual methods Computation Fluid Dynamics p.115/189

155 Iterative methods Point relaxation. Jacobi, Gauss-Seidel, SOR Line Relaxation Conjugate gradient and PCG methods Minimal residual methods Multigrid Computation Fluid Dynamics p.115/189

156 Iterative methods Point relaxation. Jacobi, Gauss-Seidel, SOR Line Relaxation Conjugate gradient and PCG methods Minimal residual methods Multigrid Fast Direct Methods (FFT) Computation Fluid Dynamics p.115/189

157 Iterative methods- Jacobi method Rewrite the discrete equations w i+1,j 2w i,j + w i 1,j ( x ) 2 + w i,j+1 2w i,j + w i,j 1 ( y ) 2 = f i,j, as 1 i N 1, 1 j M 1 Computation Fluid Dynamics p.116/189

158 Iterative methods- Jacobi method Rewrite the discrete equations w i+1,j 2w i,j + w i 1,j ( x ) 2 + w i,j+1 2w i,j + w i,j 1 ( y ) 2 = f i,j, 1 i N 1, 1 j M 1 as w i,j = 1 2(1 + β 2 ) ( wi+1,j + w i 1,j + β 2 (w i,j+1 + w i,j 1 ) ( x ) 2 with β = x / y. Computation Fluid Dynamics p.116/189

159 Iterative methods- Jacobi method This suggests the iterative scheme w new i,j = 1 2(1 + β 2 ) ( ) w old i+1,j + wi 1,j old + β 2 (wi,j+1 old + wi,j 1 old ( x ) 2 Computation Fluid Dynamics p.117/189

160 Iterative methods- Jacobi method What are suitable convergence criteria? Computation Fluid Dynamics p.118/189

161 Iterative methods- Jacobi method What are suitable convergence criteria? Suppose we write the linear system as Av = f where v is the exact solution of the linear system. If w is an approximate solution, the error e is defined by e = v w. Thus Ae = A(v w) = f Aw. Computation Fluid Dynamics p.118/189

162 Iterative methods- Jacobi method Continue iterating until residual is small enough. The residual is defined by which can be computed. r = f Aw Computation Fluid Dynamics p.119/189

163 Iterative methods- Jacobi method Continue iterating until residual is small enough. The residual is defined by r = f Aw which can be computed. For the Jacobi scheme the residual is given by r i,j = ( x ) 2 f i,j +2(1+β 2 )w i,j (w i+1,j +w i 1,j +β 2 (w i,j+1 +w i,j 1 Computation Fluid Dynamics p.119/189

164 Iterative methods- Jacobi method Therefore a suitable stopping condition might be r i,j < ɛ 1, or ri,j 2 < ɛ 2. max i,j i,j Computation Fluid Dynamics p.120/189

165 Gauss-Siedel iteration This is given by w new i,j = 1 2(1 + β 2 ) ( ) w old i+1,j + wi 1,j new + β 2 (wi,j+1 old + wi,j 1 new ( x ) 2 where the new values overwrite existing values. Computation Fluid Dynamics p.121/189

166 Relaxation and the SOR method Instead of updating the new values as indicated above, it is better to use relaxation. Here we compute w i,j = (1 ω)w old i,j + ωw i,j, where ω is called the relaxation factor, and w i,j denotes the value as computed by the Jacobi, or Gauss-Seidel scheme. ω = 1 reduces to the Jacobi or Gauss-Seidel scheme. The Gauss-Seidel scheme with ω 1 is called the method of successive overrelaxation or SOR scheme. Computation Fluid Dynamics p.122/189

167 Line relaxation The Jacobi, Gauss-Seidel and SOR schemes are called point relaxation methods. On the other hand we may compute a whole line of new values at a time, leading to the line-relaxation methods. Computation Fluid Dynamics p.123/189

168 Line Relaxation For instance suppose we write the equations as w new i+1,j 2(1+β 2 )wi,j new +wi 1,j new = β 2 (w i,j+1 +wi,j 1))+( old x ) 2 f i,j, then starting from j = 1 we may compute the values w i,j, for i = 1,..., N 1 in one go. To solve for a line we need a tridiagonal solver. Computation Fluid Dynamics p.124/189

169 Convergence properties of basic iteration schemes Consider the linear system Ax = b (-32) where A = (a i,j ) is an n n matrix, and x, b are n 1 column vectors. Computation Fluid Dynamics p.125/189

170 Convergence properties Suppose we write the matrix A in the form A = D L U where D, L, U are the diagonal matrix, lower and upper triangular parts of A, ie D = a 1,1 0 0 a 2, a 3, , a n,n Computation Fluid Dynamics p.126/189

171 Convergence properties L = a 2,1 0 0 a 3,1 a 3, a n,1 a n,2 a n,n , U = a 1,2 a 1,3... a 1,n 0 0 a 2,3... a 2,n a 3,4.. a 3,n.. 0 a n 1,n Computation Fluid Dynamics p.127/189

172 Convergence properties Then can be written as Ax = b Dx = (L + U)x + b. Computation Fluid Dynamics p.128/189

173 Convergence properties The Jacobi iteration is then defined as x (k+1) = D 1 (L + U)x (k) + D 1 b. Computation Fluid Dynamics p.129/189

174 Convergence properties The Jacobi iteration is then defined as x (k+1) = D 1 (L + U)x (k) + D 1 b. The Gauss-Seidel iteration is defined by (D L)x (k+1) = Ux (k) + b, or x (k+1) = (D L) 1 Ux (k) + (D L) 1 b. Computation Fluid Dynamics p.129/189

175 Convergence properties In general an iteration scheme may be written as x (k+1) = Px (k) + Pb where P is called the iteration matrix. Computation Fluid Dynamics p.130/189

176 Convergence properties In general an iteration scheme may be written as x (k+1) = Px (k) + Pb where P is called the iteration matrix. For the Jacobi scheme scheme we have P = P J = D 1 (L + U) Computation Fluid Dynamics p.130/189

177 Convergence properties In general an iteration scheme may be written as x (k+1) = Px (k) + Pb where P is called the iteration matrix. For the Jacobi scheme scheme we have P = P J = D 1 (L + U) For the Gauss-Seidel scheme P = P G = (D L) 1 U. Computation Fluid Dynamics p.130/189

178 Convergence properties The exact solution to the linear system satisfies x = Px + Pb. Computation Fluid Dynamics p.131/189

179 Convergence properties The exact solution to the linear system satisfies x = Px + Pb. Define the error at the kth iteration by e (k) = x x (k) then the error satisfies the equation Computation Fluid Dynamics p.131/189

180 Convergence properties The exact solution to the linear system satisfies x = Px + Pb. Define the error at the kth iteration by e (k) = x x (k) then the error satisfies the equation e (k+1) = Pe (k) = P 2 e (k 1) = P k+1 e (0). Computation Fluid Dynamics p.131/189

181 Convergence properties In order that the error diminishes as k we must have P k 0 as k, Computation Fluid Dynamics p.132/189

182 Convergence properties In order that the error diminishes as k we must have P k 0 as k, Since we see that we require P k = P k P < 1. Computation Fluid Dynamics p.132/189

183 Convergence properties From linear algebra P < 1 is equivalent to the requirement that ρ(p) = max i λ i < 1 where λ i are the eigenvalues of the matrix P. ρ(p) is called the spectral radius of P. Computation Fluid Dynamics p.133/189

184 Convergence properties Note also that for large k e (k+1) = ρ e (k). Computation Fluid Dynamics p.134/189

185 Convergence properties Note also that for large k e (k+1) = ρ e (k). How many iterations does it takes to reduce the initial error by a factor ɛ? Computation Fluid Dynamics p.134/189

186 Convergence properties Note also that for large k e (k+1) = ρ e (k). How many iterations does it takes to reduce the initial error by a factor ɛ? We need q iterations where q is the smallest value for which ρ q < ɛ giving q q d = ln ɛ ln ρ. Computation Fluid Dynamics p.134/189

187 Convergence properties Thus iteration matrices where the spectral radius is close to 1 will converge slowly. For the model problem it can be shown that for Jacobi iteration ρ = ρ(p J ) = 1 ( cos π 2 N + cos π ), M and for Gauss-Seidel ρ = ρ(p G ) = [ρ(p J )] 2. Computation Fluid Dynamics p.135/189

188 Convergence properties If we take N = M and N >> 1 then for Jacobi iteration we have q d = 2 ln ɛ ln(1 π2 2N ) = 2N ln ɛ. π2 2 Computation Fluid Dynamics p.136/189

189 Convergence properties If we take N = M and N >> 1 then for Jacobi iteration we have q d = 2 ln ɛ ln(1 π2 2N ) = 2N ln ɛ. π2 2 Gauss-Seidel converges twice as fast as Jacobi. Computation Fluid Dynamics p.136/189

190 Convergence properties For point SOR the spectral radius depends on the relaxation factor ω, but for the model problem with optimum values and N = M it can be shown that giving ρ = 1 sin π N 1 + sin π N q d = N 2π ln ɛ. Computation Fluid Dynamics p.137/189

191 Convergence properties For line SOR, and it can be shown that using optimum values ρ = ( 1 sin π 2N 1 + sin π 2N ) 2, q d = N 2 2π ln ɛ. Computation Fluid Dynamics p.138/189

192 Parabolic Equations In this section we will look at the solution of parabolic partial differential equations. The techniques introduced earlier apply equally to parabolic pdes. Computation Fluid Dynamics p.139/189

193 Parabolic Equations One of the simplest parabolic pde is the diffusion equation which in one space dimensions is u t = κ 2 u x 2. Computation Fluid Dynamics p.140/189

194 Parabolic Equations One of the simplest parabolic pde is the diffusion equation which in one space dimensions is u t = κ 2 u x 2. For two or more space dimensions we have u t = κ 2 u. In the above κ is some given constant. Computation Fluid Dynamics p.140/189

195 Parabolic Equations Another familiar set of parabolic pdes is the boundary layer equations u x + v y = 0, u t + uu x + vu y = p x + u yy, 0 = p y. Computation Fluid Dynamics p.141/189

196 Parabolic Equations With a parabolic pde we expect, in addition to boundary conditions, an initial condition at say, t = 0. Computation Fluid Dynamics p.142/189

197 Parabolic Equations Let us consider u t = u κ 2 x 2. in the region a x b. Computation Fluid Dynamics p.143/189

198 Parabolic Equations Let us consider u t = u κ 2 x 2. in the region a x b. Take a uniform mesh in x with x j = a + j x, j = 0, 1,.., N and x = (b a)/n. Computation Fluid Dynamics p.143/189

199 Parabolic Equations Let us consider u t = u κ 2 x 2. in the region a x b. Take a uniform mesh in x with x j = a + j x, j = 0, 1,.., N and x = (b a)/n. For the differencing in time we assume a constant step size t so that t = t k = k t Computation Fluid Dynamics p.143/189

200 1st order central difference approximation We may approximate our equation by [ wj k+1 wj k w k j+1 2wj k = κ + ] wk j 1. t 2 x Computation Fluid Dynamics p.144/189

201 1st order central difference approximation We may approximate our equation by [ wj k+1 wj k w k j+1 2wj k = κ + ] wk j 1. t 2 x Here wj k denotes an approximation to the exact solution u(x, t) of the pde at x = x j, t = t k. Computation Fluid Dynamics p.144/189

202 1st order central difference approximation We may approximate our equation by [ wj k+1 wj k w k j+1 2wj k = κ + ] wk j 1. t 2 x Here wj k denotes an approximation to the exact solution u(x, t) of the pde at x = x j, t = t k. The above scheme is first order in time O( t ) and second order in space O( x ) 2. Computation Fluid Dynamics p.144/189

203 1st order central difference approximation We may approximate our equation by [ wj k+1 wj k w k j+1 2wj k = κ + ] wk j 1. t 2 x Here wj k denotes an approximation to the exact solution u(x, t) of the pde at x = x j, t = t k. The above scheme is first order in time O( t ) and second order in space O( x ) 2. The scheme is explicit because the unknowns at level k + 1 can be computed directly. Computation Fluid Dynamics p.144/189

204 Parabolic pde Let us assume that we are given a suitable initial condition, and boundary conditions of the form u(a, t) = f(t), u(b, t) = g(t). Notice that there is a time lag before the effect of the boundary data is felt on the solution. Computation Fluid Dynamics p.145/189

205 Parabolic pde As we will see later this scheme is conditionally stable for β 1/2 where β = κ t. 2 x Note that β is sometimes called the Peclet or diffusion number. Computation Fluid Dynamics p.146/189

206 Fully implicit, first order A better approximation is one which makes use of an implicit scheme. Then we have Computation Fluid Dynamics p.147/189

207 Fully implicit, first order A better approximation is one which makes use of an implicit scheme. Then we have [ wj k+1 wj k w k+1 j+1 = κ ] 2wk+1 j + wj 1 k+1. t 2 x Computation Fluid Dynamics p.147/189

208 Fully implicit, first order A better approximation is one which makes use of an implicit scheme. Then we have w j k + 1 t wj k = κ k + 1 wj+1 2wj k+1 2 x + w k + 1 j 1 The unknowns at level k + 1 are coupled together and we have a set of implicit equations to solve. Computation Fluid Dynamics p.147/189

209 Fully implicit, first order Rearrange to get βwj 1 k+1 (1+2β)wk+1 j +βwj 1 k+1 = wk j, 1 j N Approximation of the boundary conditions gives w k+1 0 = f(t k+1 ), w k+1 N = g(t k+1). Computation Fluid Dynamics p.148/189

210 Fully implicit, first order The discrete equations are of tridiagonal form and thus easily solved. Computation Fluid Dynamics p.149/189

211 Fully implicit, first order The discrete equations are of tridiagonal form and thus easily solved. The scheme is unconditionally stable. Computation Fluid Dynamics p.149/189

212 Fully implicit, first order The discrete equations are of tridiagonal form and thus easily solved. The scheme is unconditionally stable. The accuracy of the above fully implicit scheme is only first order in time. We can try and improve on this with a second order scheme. Computation Fluid Dynamics p.149/189

213 Richardson method Consider w k+1 j w k 1 j 2 t = κ [ w k j+1 2w k j + wk j 1 2 x ]. Computation Fluid Dynamics p.150/189

214 Richardson method Consider w k+1 j w k 1 j 2 t = κ [ w k j+1 2w k j + wk j 1 2 x ]. This uses three time levels and has accuracy O( 2 t, 2 x). Computation Fluid Dynamics p.150/189

215 Richardson method Consider w k+1 j w k 1 j 2 t = κ [ w k j+1 2w k j + wk j 1 2 x ]. This uses three time levels and has accuracy O( 2 t, 2 x). The scheme was devised by a meteorologist and is unconditionally unstable! Computation Fluid Dynamics p.150/189

216 Du-Fort Frankel This uses the approximation w k+1 j w k 1 j 2 t = κ [ w k j+1 w k+1 j w k 1 j 2 x + w k j 1 ]. Computation Fluid Dynamics p.151/189

217 Du-Fort Frankel This uses the approximation w k+1 j w k 1 j 2 t = κ [ w k j+1 w k+1 j w k 1 j 2 x + w k j 1 ]. This has truncation error O( 2 t, 2 x, ( 4 t explicit scheme. 2 x )), and is an Computation Fluid Dynamics p.151/189

218 Du-Fort Frankel This uses the approximation w k+1 j w k 1 j 2 t = κ [ w k j+1 w k+1 j w k 1 j 2 x + w k j 1 ]. This has truncation error O( 2 t, 2 x, ( 4 t explicit scheme. 2 x )), and is an The scheme is unconditionally stable, but is inconsistent if t 0, x 0 but with t / x remaining fixed. Computation Fluid Dynamics p.151/189

219 Crank-Nicolson A popular scheme is the Crank-Nicolson scheme given by w k+1 j w k j t = κ 2 " w k+1 j+1 2wk+1 j + w k+1 j 1 2 x + wk j+1 2wk j + wk j 1 2 x #. Computation Fluid Dynamics p.152/189

220 Crank-Nicolson A popular scheme is the Crank-Nicolson scheme given by w k+1 j w k j t = κ 2 " w k+1 j+1 2wk+1 j + w k+1 j 1 2 x + wk j+1 2wk j + wk j 1 2 x #. This is second order accurate O( 2 t, 2 x) and is unconditionally stable. (Taking very large time steps can however cause problems). As can be seen it is also an implicit scheme. Computation Fluid Dynamics p.152/189