Finite Difference Method for PDE. Y V S S Sanyasiraju Professor, Department of Mathematics IIT Madras, Chennai 36

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1 Finite Difference Method for PDE Y V S S Sanyasiraju Professor, Department of Mathematics IIT Madras, Chennai 36 1

2 Classification of the Partial Differential Equations Consider a scalar second order partial differential equation (PDE) in d independent variables, given by LT a b cs t x x x d where, s 0, T,, d (4.1.1) Unless otherwise stated, repeated index stands for summation The stationary form of the Eq. (4.1.1) is given by LS a b cs x x x (4.1.)

3 Equation (4.1.) is obtained after Renaming b -a as b α and also using a ba,x a x x x x Further, the coefficients of the cross derivates are made equal by enforcing a αβ = a βα =1/(a αβ + a βα ). Arranging the coefficients a αβ (real) in a matrix form, say, A, gives a symmetric matrix of size (d x d) with real coefficients. Such a matrix will have d real eigenvalues and the corresponding eigenvectors are linearly independent. 3

4 Classification Differential equation (4.1.) is called Elliptic if the eigenvalues of A are non-zero and have the same sign Hyperbolic if only one eigenvalue has sign different from all others Parabolic if precisely one eigenvalue is zero, while the other have the same sign and rank of (A, b) is equal to the dimension of the problem with the vector b has the elements of b α. For the elliptic case, A is definite and can be treated as positive definite. If not, the equation can be multiplied with a minus one (-1) and can be made to Positive definite. 4

5 Tensor form of second order partial differential equation, given in (4.1.1) and (4.1.), is good for classifying a PDE in d independent variables. However, for d =, it is easier to use the quasi-linear form of equation (4.1.), in its explicit form, given by x x y y x y A( x, y) B( x, y) C( x, y) D E F 0 (4.13) where A, B and C are functions of the independent variables x and y alone. Then the classification of the PDE (4.1.3) depends on the sign of the discriminant (B -4AC). Notice that the coefficient of the first derivative terms don t contribute to the classifications 5

6 Equation (4.1.3) is hyperbolic in the regions where the discriminant is positive and is elliptic when it is negative. The equation is called parabolic whenever (B -4AC) is zero. Well known examples for elliptic, hyperbolic and parabolic PDE are : Poisson s equation Wave equation f ( xy, ) x y t x c and Unsteady diffusion equation t K x, respectively. 6

7 Classification of Non Stationary Equations For the transient case, a first derivative with respect to time is an additional term when compared to the corresponding stationary differential equation. Since the first derivatives does not contribute to the classification, the existing elements of A are same for both stationary and transient cases. However, due to the increase of the number of independent variables by one (time variable t), the number of rows and columns of A will increase by one with zero elements. This will result into an additional zero eigenvalue. 7

8 Therefore, from the classification in terms eigenvalues, if the stationary equation is elliptic then the corresponding unsteady equation is parabolic. The same in equation form can be written as As 0 bs L S s A T and b t where A T and A S are the matrices with coefficients of the second derivative terms of transient and stationary equations, respectively. 8

9 Initial and Boundary Conditions Unsteady equations require initial conditions given by 0 0, x ( x), x, t 0 (4.1.4a) And the boundary conditions, on Ω, (0 < t < T), which can be any one of the following: 1. Dirichlet t, x f t, x. Neumann K t, x f t, x n (4.1.4b) 3. Robin K tx, a tx, f tx, n 9

10 Indirectly boundary conditions play a very important role in the classification of the PDE. For elliptic equations, the boundary conditions, which have to be prescribed all along the boundary of the domain, influence the solution everywhere. For hyperbolic equations, a local change in the boundary data influences only a part of the domain resulting into a domain of influence and domain of dependence. The CFL condition of the numerical schemes used in CFD is based on this principle. Finally, for parabolic equations, a time like variable can be identified and any changes in the boundary data influences the entire domain however only at the later times. 10

11 Physical Significance of the Classification To understand the significance of the classification of the PDE, one should look at the physical nature of the solutions produced by these equations. To make the computations easier, fix all the coefficients, except a αβ in (4.1.) to be zero. The simplified equation is then given by (4.1.5) x a x Assume a plane wave solution, given by 0 iw( x) e, w( x) n. x b x where w( ) describes the wave fronts or characteristic surfaces 11 given by w( ) = constant x

12 Substituting the wave solution in (4.1.5) gives a n n 0 with x grad w T or n A n 0 Since A is symmetric and real, eigenvalues λ α and the ( corresponding orthogonal eigenvectors ), 1,...d are all real. ( ) Fixing n c T (since ( ) 1), gives ( ) T T ( ) ( ) c A c c 0 If (4.1.5) is elliptic then, since all the eigenvalues are positive, therefore no possibility of λ α c α (summation due to repeated index) becoming zero. This contradicts the existence of any wave like solution in this case. If (4.1.5) is hyperbolic, then the (single) negative eigenvalue can be expressed in terms of the other eigenvalues and coefficients c α. 1

13 That shows the existence of wave like solution for hyperbolic type equations. Finally, if (4.1.5) is parabolic then exactly one eigenvalue is zero, therefore one can fix the corresponding c α as non-zero and all the other coefficients to be zero. That is, one particular direction can be identified in which wave like solution can exists. To conclude, there is no possibility of wave like solutions to exist for elliptic equations, however a clear case of existence of wave like solutions, in any of the many directions, for the hyperbolic equations. And finally, existence of exactly one particular direction in which wavelike solutions may form for parabolic equations. 13

14 Convection Diffusion Equation (CDE) A general convection diffusion equation is given by u s, x t T, 0, 1,,..., d t x x x (4.1.13) where ε is the constant diffusion parameter. For small values of ε, Eq. (4.1.13) has layer solutions (solution varies very rapidly in a small region(s) particularly towards any physical boundary of the problem). The study of Eq. (4.1.13), when the diffusion parameter is small, is called the boundary layer theory or singular perturbation theory. 14

15 The stationary form of the convection-diffusion equation is given by u s (4.1.14) x x x The convection diffusion equation which is also known as Burger s equation is similar to some of the governing equations of the fluid dynamics but without any pressure term. Further, due to the layer behavior of its solutions obtaining accurate numerical solutions is a challenge 15

16 Non Dimensionlization It is better to non-dimensionalize the governing equations representing any physical phenomena before they are solved numerically. If L and V are the characteristic length and velocity components, respectively, then the non-dimensionalization of the variables in (4.1.14) can be done using ' ' u ' tv ' x ' sl u t x s V L L V (4.1.15) 16

17 Using the non dimensional variables (4.1.15), equation (4.1.14) ( can be written as (after dropping the symbol VL 1 u Pe x x x (4.1.16) where Pe is the Peclet number which gives the ratio of the convection and diffusion coefficients. Large values of Peclet represent the dominance of convection process and small values of Pe represent the dominance of the diffusion process. In general, Eq. (4.1.16) is elliptic but behaves like an hyperbolic equation for large values of Pe. The corresponding transient equation is parabolic in its behavior. s 17

18 Discretization There are three steps in numerically solving the differential equations: 1. Discretization of the domain by placing a large number of nodes with help of grid generation techniques.. Approximating (or discretizing) the governing equations at the nodes identified in Step Solving the algebraic equations obtained in Step using direct or iterative methods. 18

19 For regular domains placing the nodal points uniformly and connecting them by straight lines gives the required grid. For example, the discretization of a one dimensional domain that is, an interval, can be realized as follows: Fig Discretization of an interval 19

20 In this discretization, a set of uniformly distributed points x 0, x 1,.x n are identified such that x 0 =a,x n =band x i x i-1 = x for i=1,, n. where, x is the step length. A uniform grid means the distance between any two consecutive points x i and x i-1 is constant. Otherwise, the discretization is called non-uniform. The coordinate of any mesh point is computed using xi a i* x i 0,1,... n Function value u at any point x i is represented by 0,1,... u xi ui i n (4..1) 0

21 If Ω is a rectangular two-dimension domain bounded by [a,b] x [c,d], then it can be discretized as: Fig. 4.. Discretization of a rectangular domain The coordinates of any point p(x i,y j ) are obtained using x i x 0 i * x, i 0,1,,..., n, y j y 0 j * y, j 0,1,,..., m (4..) 1

22 Further, the dependent variable u at any point P is represented using u i,j =u(x i,y j ). A similar extension can be carried out for higher dimensional domains. The second step is the discretization of the governing equations. To realize this, there are several methods. In the present lecture, the Taylor series based method is highlighted. Consider a function u which depends on the independent variable x in the interval [a, b].

23 Let the function u be sufficiently smooth (differentiable) and it has values u i and u i+1 at any two neighboring points, i and i+1 respectively. By using Taylor series expansion, u i+1 can be expressed in terms of u i and its higher derivatives as x x 3 ui1 uxi x ui xui ui ui...! 3! where the superscript stands for a derivative with respect to x. (4..3) From equation (4..3), u i can be written as u i 1 ui x x ui ui ui... x! 3! ui 1 ui 1 O( x) ui O( x) x x (4..4) 3

24 where δ + is the forward difference operator defined by δ + u i = u i+1 u i Equation (4..4) is the first approximation to u at the node x i. In this approximation x x ui ui...! 3! is the error. This error is called Truncation error, in which leading term. x ui! is the Since the degree of the step size is one in the leading term of the. u error, (4..4) is a first order approximation for 4

25 The order of approximation is an important concept in the process of discretization which gives an immediate insight about what kind of accuracy can be expected from the scheme. Similarly, one can also write x x 3 uxi1 ui1 ui xui ui ui...! 3! ' u i u i 1 x x 1 ui ui ui... ui O( x) x! 3! x (4..5) (4..6) where δ - is the backward difference operator defined by δ - u i =u i u i-1. Eq. (4..5) is again first order as the degree of the step size in the leading term of the error is also one. 5

26 From (4..5) and (4..4) (subtracting (4..5) from (4..4)) one can write x 3 x 5 u 1 1 v i ui xui ui u... 3! 5! i (4..7) 1 x 1 u 0 ( i ui1 ui1 ui ui O x ) x 6 x (4..8) Here, δ 0 is called the central difference operator defined by δ 0 u i =u i+1 u i-1. u Equation (4..8) is a second order accurate approximation for as the leading term of the error has x in second degree. 6

27 Alternatively, adding (4..4) and (4..5) gives x x 4 u 1 1 iv i ui ui ui u...! 4! i ui 1ui ui 1 x iv 1 i i i u u u O( x ) x 1 x (4..9) (4..10) where δ is the central difference operator for second derivative which is a second order accurate approximation. 7

28 Numerical Implementation The finite difference approximations (4..4), (4..6), (4..8), and (4..10) can be used to replace the first and second order derivative terms of a differential equation to convert it into a difference (possibly linear algebraic) equation at every nodal point of the interior of the domain. Note that the central approximation (4..8) is one order better accurate than the forward (4..4) and backward (4..6) approximations. To understand the implementation, consider the following boundary value problem (BVP). d u sin(ln ) 0, 1, (1) 1and () du u x x u u dx xdx x (4..11) 8

29 The analytical solution of the BVP Eq. (4..11) is x ux ( ) x 3sin log( x) 5cos log x x 34 (4..1) The finite difference solution of Eq. (4..11) is obtained by replacing its first and second order derivative terms of with (4..7) and (4..8), respectively to get ui1 ui ui1 ui1 ui1 O( x ) u sinlog( x ) 0 i i x xi x x i for i 1,,..., n1 (4..13) Note that at node numbers 0 and n, we have boundary conditions and the given differential equation may not be valid at these points, therefore, at these nodes we have (from the boundary conditions) 9 u 0 = 1.0 and u n =.0 (4..14)

30 Equation (4..13) has n-1 equations in n+1 variables, therefore adding Eq. (4..14) closes the system to solve for the unknowns u i,fori=1,,...,n-1. Rearranging the terms in (4..13) gives sin log( ) u i u u x ( ) i i i x xi x x x x xi x i i 1,,... n1 (4..15) Equation (4..15) is a linear system with tri-diagonal coefficient matrix. Solving such linear algebraic systems is the third and final step of the numerical schemes. 30

31 After solving Eq. (4..15) with step lengths ( x) 1/40, 1/80 and 1/160, the absolute errors in the numerical solution are computed and compared in the Fig Fig Comparison absolute errors with three distinct discretizations 31

32 Higher Order Approximations Consider the operators: u u Du D u Eu u x x n i n i i, i, n i i1 (4.3.1) u x n u x i n i Let Du,, be the first and n th i D ui Eu n i ui 1 order derivative and shifting operators. Using the operators (4.3.1) in (4..3) gives xd 1 1 E e, D log( E) D log(1 ) x x (4.3.) 3

33 Expanding the log function gives Du i ( ) ( ) ( )... x 3 4 u i (4.3.3) Similar replacements of E with backward difference operator δ - or central difference operator δ gives Du i ( ) ( ) ( )... x 3 4 u i (4.3.4) Du i x u i (4.3.5) where ui ui 1/ ui 1/ 33

34 Similarly, using the same procedure, the formulae for second derivative can be obtained as Du x x i x (4.3.6) Approximations (4.3.3), (4.3.4), (4.3.5) or (4.3.6) can be used to generate higher order approximations. However, as we increase the order of approximation, the number of terms in the approximation is also increase andsuchformulae may not be convenient to use in the approximation of PDE at the points particularly close to the physical boundaries. 34

35 Numerical Illustration The central difference approximation in (4.3.6) may be used until second term to generate a fourth order approximate formula for the Poisson equation (in two dimensions) u u f ( xy, ) x y Potential flow equations, discussed in Module have a similar form. Consider the required fourth order approximations of the second derivatives in x and y directions, respectively, at any typical nodal point, say (i, j), as, u 1 1 u 1 1 x x 1 y y 1 1 x x ui, j and 1 y y ui, j 35

36 Then the fourth order approximation of the Poisson equation u u at any nodal point (i, j), is given by f ( xy, ) x y Since u x 1 y 1 x x y y i, j i, j u and x x 1 x x ui, j u y y 1 y y ui, j f 36

37 For the sake of simplicity assume, x = y (the following procedure is also valid without this assumption). Rewrite the discretized equation as u x f 1 1 x x y y i, j i, j Multiplying with 1 x 1 y 1 1 gives (after using the commutative nature of the finite difference operators) x 1 y y 1 x ui, j x 1 y fi, j x

38 Expanding the terms using 1 1 x y 1 1 1, on the left hand side 1 x x x y y y and neglecting the terms which are having the order more than two gives u x 1 1 f x y y x i, j x y i, j 38

39 1 1 u x 1 f x y x y i, j x y x y i, j Neglecting the product term (since it produces terms at higher order level) on the right hand side, the final scheme can be written as 1 1 u x 1 f 6 1 or x y x y i, j x y i, j u x f x y x y i, j x y i, j 39

40 Expanding the central difference operator in the above scheme gives u i1, j ui 1, j ui, j1 ui, j1 ui 1, j1 ui 1, j1 ui 1, j1 ui 1, j1 8 40u x 8 f f f f f ij, ij, i1, j i1, j ij, 1 ij, 1 (4.3.7) Since f is a known function, the right hand side of Eq. (4.3.7) is known at each nodal point (i, j) and its surrounding points. Finally, the Taylor series expansion of Eq. (4.3.7) demonstrates that it is fourth order accurate in both x and y. 40

41 Higher Order By Computing the Coefficients of a Stencil Another approach to generate higher order approximations is to fix a suitably large stencil and then computing the coefficients through Taylor series expansion and comparing the coefficients. To comprehend this, consider the differential operator A B x x (by appropriately fixing the values of A and B, formulae for a particular derivative also can be obtained) and equate this to au 0 i au 1 i1au i1au 3 i au 4 i,thatis u u u u A B a u au a u a u a u x x 0 i 1 i1 i1 3 i 4 i (4.3.8) 41

42 Expanding the terms on the right hand side using Taylor series and simplifying gives u u u A B a a a a a u a a a a x x x x i, j ( ) ( ) i, j x ui, j x ui, j ( a a 4a 4 a ) ( a a 8a 8 a ) x 6 x x u u ( a a 16a 16 a ) ( a a 3a 3 a ) 4 x 10 x i, j x i, j Comparing the coefficients of Eq. (4.3.9) gives A a a a a a 0, a a a a, x B a a 4 a 4a, a a 8a 8a 0, a a 16a 16a x (4.3.9) (4.3.10) 4

43 Finite Difference Schemes for Elliptic Equations Consider a two-dimensional Poisson equation given by u u f( x, y), ( x, y) x y (4.4.1) First, generate the grid by discretizing the domain Ω with step lengths x and y in x and y directions, respectively (for an example in a rectangular domain, refer Fig. 4..). Then at each grid point of the domain, discretize the given Eq. (4.4.1) by replacing the partial derivatives of the equation with second order finite difference approximations (4..10) to get 43

44 ui1, j ui, jui1, j ( ui, j1 ui, jui, j1 O x ) O( y ) f i, j x y for i 1,,..., nx 1, j 1,,..., ny 1 (4.4.) where n x and n y are the number of grid points used to discretize the domain in x and y directions, respectively and f i,j is the source function at the point (x i,y j ). Equation (4.4.) is a closed algebraic system, if Eq. (4.4.1) is supplemented with Dirichlet boundary conditions. On the other hand if the given boundary conditions are mixed type (Neumann or Robin) like u a bu c (4.4.3) n where n is the normal to the boundary, then to close the system 44

45 The derivative terms in Eq. (4.4.3) are also approximated using the first order accurate forward or backward difference approximations at every grid point of the boundary, that is, at i = 0orn x or j =0orn y. Particularly, if the boundary condition Eq. (4.4.3) is given at i = 0 or at j = 0 then the forward difference approximation and if it is given at i = n x or at j = n y then backward difference approximations maybe used for the discretization. Since the forward or backward difference approximations are only first order accurate, the overall accuracy of the problem is considered as first order though the governing equation is approximated to the second order accuracy. 45

46 Finally, the algebraically closed system is solved using any linear solver Alternatively, to raise the overall order of accuracy to two, Insert ghost nodes as shown in the Fig outside the domain Discretise the governing equation also at the boundary points over which the derivative boundary conditions are prescribed. Fig : Discretization with ghost points 46

47 Then eliminate the data points at these ghost points using the equations obtained by approximating the derivative boundary conditions using second order central difference approximations. Solver for (4.4.) : Due to the sparseness of the coefficient matrix generated through (4.4.) (note that only five diagonals of this matrix, whatever the size of the system, has non-zero entries) using any direct method unnecessarily increases the number of computations. Further, the iterative methods like Jacobi, Gauss Siedel are too slow in their convergence, therefore, alternatively one can apply the ADI (Alternate Direction Implicit) method which is based on line-gauss Seidel solver. 47

48 Implementation of the ADI method In the first step, on each vertical line, the tri diagonal system u k , u k, u k f k k ij 1, ij, i j i j u ij, 1 u ij, 1 y x x y x y is solved (n y -1 times) using Thomas algorithm (refer Module 5) In the second step, on each horizontal line for i 1,,..., n x u k1 1 1, 1 u k, u k f k k ij, 1 i, j i j i j u i1, j u i1, j y x y y x for j 1,,..., n y 1 is solved (n x -1 times) once again using Thomas algorithm The procedure is repeated until the convergence after starting the iterative process with iteration number as 0. 48

49 Numerical Illustration Consider the two-dimensional heat flow, governed by u x u y u 0 (4.4.5) where is the Laplace operator, in a rectangular duct. The duct surfaces are considered to be perfectly insulated. The lengths are (x,y) = (4, 3) with boundary conditions and discretization as shown in the Fig The objective is to find the temperature distribution on the surface of the sheet. 49

50 Figure 4.4. Domain and boundary conditions Solution: With the discretization of the domain as given in the Fig. 4.4., the step lengths are in x and y directions, respectively. 1,

51 The discretization of the given Laplace equation using second order central difference approximations is given by u u u u u u i1, j i, j i1, j i, j1 i, j i, j1 0 ( where i 1,, 3 & j 1, ) x y i1, j i, j i1, j i, j1 i, j i, j1 4 u u u 9 u u u 0( where i1,,3& j1,) (4.4.6) (4.4.7) Solving the six equations in the (4.4.7), using any direct method, gives u u u ,1 1, ,1, u u u 3,1 3, 51

52 Note: 1. If Gauss-Seidel iterative method with stopping criteria u - u 10 ( k + 1) ( k) - 5 where k is the iteration number, and the initial approximation to the solution as zero are used then the scheme converges in 85 iterations.. If the Gauss-Seidel method is replaced with SOR (with relaxation parameter 1.1), then the scheme converges in 55 iterations for the same conditions used in the above. 3. The analytical solution of the problem is , ,

53 Note: 4. The percentage errors in the obtained numerical solution are 54%, 30%, 9% which are very high. 5. If the step lengths are changed to /40 and 1/30 in x and y directions, respectively then the solution is improved to 0.760, 3.818, and the percentage errors are then reduced to 0.5%, 0.3%, 0.078%. 53

54 Laplace Equation in Circular Geometries Consider the steady diffusion problem over a thin circular plate which is governed by Laplace equation in polar coordinates given by, u 1u 1 u 0 0 rr, 0 a r r r r (4.4.8) where r and θ are radial and angular coordinates. 54

55 If Dirichlet conditions are given at r = 0 and r a, and periodic conditions are used in angular direction then the second order central difference approximation of (4.38) gives ui 1, jui, jui 1, j ui 1, jui 1, j ui, j 1 ui, jui, j 10 dr r dr r d i i (4.4.9) where dr and dθ are step lengths in radial directions, respectively. and angular Varying i and j in Eq. (4.4.9) gives a linear system which can be solved for the solution of Eq. (4.4.8). 55

56 In the absence of any boundary condition at r = 0, discretization of Eq. (4.4.8) in the conventional way leads to one over zero in second and third terms of Eq. (4.4.9). In such cases, u at r =0is obtained using u 0 1 n u i n 1 i0 rr a (4.4.10) To obtain Eq. (4.4.10), discretize the Cartesian equivalent of Eq. (4.4.8) at the center of the circle with radius r a and take the mean after repeating the same on the (n θ +1 times) rotated stencil. 56

57 Difference Schemes for Parabolic Equations One-dimensional problems: Consider the unsteady diffusion problem (parabolic in nature) in a thin wire governed by the differential equation u t u k, x ( a, b ), t 0 x (4.5.1) Assume that the initial conditions, the distribution of u at t =0 and the boundary conditions, u at x = a and b are given. 57

58 Forward time central space (FTCS) scheme A simple and easiest scheme to compute the numerical solution of (4.5.1) is the FTCS (forward time and central space) scheme which is an explicit method. An explicit scheme uses a stencil in which only one unknown is written in terms of the remaining known values at other stencil points. The FTCS approximation of Eq. (4.5.1) is 1 n1 n 1 n n n ui ui k ui1ui ui 1 t x n1 n n n n t ui ui rui 1 ui ui 1, r k x u ru (1 r) u ru i 1,,, n 1, n0,1, n1 n n n i i1 i i1 x (4.5.) where, the superscript n represents the time level. The discretization and the stencil of the FTCS Eq. (4.5.) is shown in the Fig

59 Fig Discretization and the stencil of FTCS scheme In the Fig , there is only one stencil point at n+1 th time level which is the unknown and three points at n th time level which anyway are known. Therefore using Eq. (4.5.), one unknown at a time at n+1 time level can be computed by varying i = 1,,..., n x-1. 59

60 Taylor series expansion of Eq. (4.5.) demonstrates that the FTCS scheme is first order accurate in time and second order in space. Backward time central space (BTCS) scheme The BTCS approximation of Eq. (4.5.1) gives 1 n n1 1 n n n ui ui k ui1ui ui 1 t x n n n n1 t rui 1(1 r) ui rui 1ui, r k, i 1,,, n 1, 1,, x n x (4.5.3) Equation (4.5.3) is an implicit scheme which has more than one unknown at n th time level, therefore one equation alone can t be solved unless it is clubbed with more number of equations to close the system. 60

61 This is done by grouping all the discretized equations at a particular time level and solving them in one step using say, Thomas algorithm if the resultant system is a tri-diagonal one. The same is repeated by incrementing the value of n until the required time level is reached. This type of procedure is called a time marching scheme. 61

62 The computational stencil for the BTCS scheme is as shown in Fig Fig Fig Computational stencil for BTCS scheme At each time step, the scheme Eq. (4.5.3) also can be written as n1 n1 n1 n t rui 1 (1 r) ui rui 1 ui, r k, i 1,,, n 1, 0,1, x n x (4.5.4) 6

63 Weighted average scheme Weighted average of Eqs. (4.5.1) and (4.5.3), gives 1 n1 n k n n n n1 n1 n1 ui ui (1 ) ui1ui ui 1ui1 ui ui 1 t x ru (1 r) u ru (1 ) ru (1 (1 ) r) u (1 ) ru n1 n1 n1 n n n i1 i i1 i1 i i1 t r k, 0 1, i 1,,, n, 0,1, x n x (4.5.4) For θ equals to 0 and 1, Eq. (4.5.4) gives FTCS and BTCS schemes, respectively. The Taylor series expansion of (4.5.4) gives n 3 n 4 n 4 n u u 1 ui t ui k x ui k t x ui k t (4.5.4) t x t t 1 x t x 3 4 Therefore, weighted average scheme Eq. (4.5.4) is second order 1 accurate in space and first order in time if. The scheme is 1 also second order accurate in time if. The weighted average scheme with is known as Crank-Nicolson scheme. 63 1

64 u Numerical Illustration Consider, x (0,1), t with initial conditions u(x,0) = sin t x 0 x and boundary conditions zero at x = 0 and 1. Use step sizes 0. and 0.01 in x and t directions, respectively, Compare, after ten time steps, the numerical solutions obtained with FTCS, BTCS and Crank-Nicolson schemes with the t analytical solution e sin x. For the step size 0., we have u kt 1* 0.01 r 0.3 x 0.*0. With r = 0.3, the FTCS, BTCS and Crank-Nicolson schemes are given by 64

65 u 0.3u 0.4u 0.3u n1 n n n i i1 i i1 0.3u 1.6u 0.3u u n1 n1 n1 n i1 i i1 i 0.15u 1.3u 0.15u 0.15u 0.7u 0.15u n1 n1 n1 n n n i1 i i1 i1 i i1 for i 1,,3,4, n0,1,,,9 (4.5.6) Equation (4.5.6) depends only on the value of r and is independent of the step sizes. The solution and the percentage errors generated by the three schemes (FTCS, BTCS and Crank_Nicolson), after marching 10 times in the time direction using Eq. (4.5.6), are compared in the Table

66 X Analytical FTCS BTCS Crank-Nicolson Solution Error Solution Error Solution Error % % % % % % Table Comparison of the analytical and numerical solutions and their errors The initial and boundary conditions in the above computations are taken from the exact solution. 66

67 Two-dimensional problems Consider the unsteady diffusion over a flat plate governed by the differential equation u u u (4.5.7) k, ( x, y) ( a, b) X( c, d), t 0 t x y Assume that the initial and boundary conditions on u are known. Explicit scheme Approximating the time derivative with forward difference and space derivatives with central differences gives a scheme 1 n1 n 1 n n n 1 n n n ui, j ui, jk ui1, jui, jui1, j ui, j1 ui, jui, j1 t x y u u r u u u r u u u n1 n n n n n n n i, j i, j 1 i 1, j i, j i 1, j i, j 1 i, j i, j 1 t t r k, r k x y n1 n n n n n ui, j ru i, j1ru 1 i1, j(1 r1 r) ui, j ru 1 i1, jru i, j1 i 1,,, n 1, j 1,,, n 1, n 0,1, x 1 y (4.5.8) 67

68 Here, y is the step length in y direction. Taylor series expansion of Eq. (4.5.8) shows that the explicit scheme is first order accurate in time and second order in space (both in x and y directions). Weighted average or Crank-Nicolson type of approximation to (4.5.7) gives a penta-diagonal system like Eq. (4.4.) at the n+1 th time level, solving such a system is very expensive computationally, therefore, alternatively ADI method can be developed as follows: u u u u u u u u t/ x y n n n n n n n n i, j i, j i1, j i, j i1, j i, j1 i, j i, j1 k u u u u u u u u t/ x y n n n n n1 n1 n1 n1 i, j i, j i1, j i, j i1, j i, j1 i, j i, j1 k (4.5.9) 68

69 First on j is constant lines, using the first tri-diagonal part of Eq. (4.5.9), solution at n+1/ time level is obtained. In the second step, using the solution at the n+1/ time level over the i is constant lines and using second tri-diagonal part of Eq. (4.5.9), the solution is marched to the n+1 time level. Therefore, for each time level, Eq. (4.5.9) gives (n x -1 + n y -1) tridiagonal systems in x and y directions which are mush easier to solve using Thomas algorithm than the penta-diagonal system (of size L X L, where L = (n x -1) * (n y -1)) appears in the weighted average or Crank-Nicolson schemes for two dimensional problems. 69

70 Convergence Here, convergence means, the convergence of the numerical solution to the analytical solution For parabolic and also for all time dependent problems, the convergence of the numerical solution to the corresponding analytical solution is carried out through the testing for consistency and stability since, according to Lax equivalence theorem, Consistency and stability are necessary and sufficient conditions for convergence of the finite difference solutions of any time dependent problem. 70

71 Consistency of a Numerical Scheme Under the limiting case of step lengths tending to zero, if a finite difference scheme converge to the corresponding differential equation then such a scheme is called consistent. Mathematically, it is tested by looking at the truncation error of the scheme as the step lengths tend to zero. If the truncation error tends to zero as the step lengths tend zero then the numerical scheme is said to be consistent. 71

72 Numerical Illustration Consider the Taylor series expansion Eq. (4.5.5) of the Weighted average scheme Eq. (4.5.4) ru (1 ru ) ru (1 ) ru (1 (1 ) ru ) (1 ) ru given by n1 n1 n1 n n n i1 i i1 i1 i i1 n 3 n 4 n 4 n u u 1 ui t ui k x ui k t x ui k t 3 4 t x t t 1 x t x Taking step lengths x and t tending to zero, the series expansion converges to the governing equation k, therefore, the weighted average scheme is consistent. u t u x 7

73 Stability of a Numerical Scheme Stability of a numerical scheme deals with the growth of the rounding errors during the time marching process. Let us illustrate, the stability through the following observation Repeat the computations of Eq. (4.5.6) once again with time steps and 0.0 (that is, for the value of r as 0.45 and 0.55, respectively) for FTCS and Crank-Nicolson (CN) schemes The corresponding solution with r =.45 and.55 are presented in Tables 4.5. and 4.5.3, respectively 73

74 X Analytical FTCS Crank-Nicolson Solution Error Solution Error % % % % Table 4.5. Comparison of the solution and errors with r = X Analytical FTCS Crank-Nicolson Solution Error Solution Error % % % % Table Comparison of the solution and errors with r =

75 Its clear from the Tables 4.5. and that, with r = 0.45, the FTCS scheme produces solutions with errors less than.% and CN scheme with errors less than 1.3%. However, if the value of r is increased to 0.55, the CN scheme continues to give solutions with similar errors while the errors in FTCS scheme increased by many folds. The behavior of increasing errors with FTCS becomes worse and completely dominated by these errors if we still continue the computations for higher time levels. This is due to the instable nature of the FTCS scheme when the value of r is greater than 0.5. Mathematically this can be understood using the following analysis: 75

76 n Let (u) = 0 be a linear difference scheme and is its numerical n n solution, U is the exact solution and is its error at n th i E i time level at the nodal point x i then we have u n n n (4.5.10) i Ui Ei and n n n n n n ( ui ) = ( U E ) = ( ) + ( E i ) = ( E i ) = 0 (4.5.11) i i U i That is, the error also satisfies the same difference equation which numerical solution satisfies. Therefore, one can study the behavior of the error by studying the numerical solution itself. u i 76

77 Further, if the numerical solution is assumed to be periodic, achieved by reflecting the solution in the region (0, L) in to (-L, L) and expressible in terms of finite Fourier series (since the domain is of finite length which is discretized with finite step length) then it can be written as N N N n n Ik jxi n Ik ji x n Ii (4.5.1) u K e K e K e i j j j jn jn jn where N is the number of points in the discretization, (N=L/ x) I 1 n K is the amplitude of the j th j harmonic, k j is the wave number is the phase angle given by k j x. 77

78 Note: k j varies from N to N instead of - to, because the maximum and minimum resolvable wavelengths (λ) are only L and x, respectively and the maximum wavelength L is discretized with N+1 points, that is from N to N. In the actual computation, due to the linear nature of the difference scheme, it is enough to use one Fourier mode, instead of (4.5.1) and looking at the raise or damping of the amplification factor G which is defined as the ratio of the amplitude at n+1 and n th time levels, that G is defined as n1 Ki G (4.5.13) n K Any scheme (linear) is said to be stable if G < 1 (4.5.1) Otherwise is said to be unstable. i 78

79 Numerical Example: Discuss the stability criteria of the weighted average scheme ru (1 r) u ru (1 ) ru (1 (1 ) r) u (1 ) ru n1 n1 n1 n n n i1 i i1 i1 i i1 (4.5.15) n n Ii Substituting u K e in Eq. (4.5.15) and simplifying for i amplification factor G, gives n1 I I K (1 ) re (1 (1 ) r) (1 ) re 14(1 ) r sin G n I I K re (1 r) re 14r sin (4.5.16) For θ = 0, that is, for FTCS scheme, G <1 implies whichistruewheneverthevalueofr<1/. 1 4r sin 1 79

80 Therefore, FTCS scheme is only conditionally stable. However, for θ greater or equal to ½, Eq. (4.5.16) is unconditionally stable. Once again looking at the Tables 4.5. and 4.5.3, it is clear that forr=0.45,ftcsschemeisabletoproduceaccuratesolutions but the round of errors are dominated when the value of r is raised to 0.55 because FTCS scheme is not stable at r = On the other hand due its stable nature of CN scheme for all values of r, the round of errors are under control at both r = 0.45 and

81 Difference schemes for hyperbolic equations Consider the one dimensional wave equation u t u c, x(0, L), t 0 x (4.6.1) where c is a positive constant. Equation (4.6.1) requires two initial conditions at t=0 u(x,0) = f(x), u t (x,0) = g(x) (4.6.) and two boundary conditions at x=0 and L given by u(0,t) = u(l,t) = 0 (4.6.3) 81

82 Explicit Scheme Discretizing the second derivatives in Eq. (4.6.1) with central differences gives n1 n n1 n n n ui ui ui ui 1ui ui 1 O( t ) c O( x ) t x i 1,,3,, n 1, n 1,,3, 1 1 u u u u u u O( t, x ) n1 n 1 n n n n i i i i i i u u (1 ) u u O( t, x ) n1 n n n i i1 i i1 ct, i 1,,3,, nx 1, n 1,,3, x x (4.6.4) (4.6.5) For n=1, Eq. (4.6.5) requires the solution at t =- t, which can be obtained, using the derivative boundary condition at t=0 Eq. (4.6.). 8

83 Discretizing the initial condition: At n=1, Discretising the derivative initial condition with central difference gives u u t 1 1 i i 1 1 ( ) ( i) i i i ( ) O t g x u u tg O t (4.6.6) Eqs. (4.6.5) with (4.6.6) at n=1, completes the discretization and gives a second order solution in both space and time. 83

84 Alternatively, using Taylor series at n=1, one can write 3 t t ux ( i, t1) ux ( i,0 t) ux ( i,0) tut ( xi,0) utt ( xi,0) uttt ( xi,0)! 3! t fi 1 fi fi 1 3 fi tgi O( x, t ) using(4.6.)! x (4.6.7) Therefore, Eq. (4.6.7) may be used to compute the solution at t= t and as usual Eq. (4.6.5) at all later time levels. Since, from the initial condition Eq. (4.6.), the function f is known explicitly, the second partial derivative of f with respect to x in Eq. (4.6.7) can be replaced with exact partial derivative. 84

85 Consistency and Stability of Eq. (4.6.7) Taylor series expansion of Eq. (4.6.5) gives u u 1 u u t x 1 t x c t c x O t x 4 4 (, ) (4.6.8) Equation (4.6.5) converges to the governing Eq. (4.6.3) if the step lengths x, t tending zero, therefore the scheme Eq. (4.6.5) is consistent. Equation (4.6.8) also indicates that the scheme Eq. (4.6.5) is second order in both t and x. 85

86 For verifying stability, u K e n n Ii i in Eq. (4.6.5) and simplify to get K e K e Ke (1 ) Ke Ke n1 Ii n1 Ii n I( i1) n I( i) n I( i1) Dividing with n Ii Ke gives 1 I( 1) n I(1) 1 KK e (1 ) K e K cos (1 ) K (1 sin ) (1 ) K10 K 1sin K10 K AK1 0 with A1sin Computing K from Eq. (4.6.9) gives, K A A 1 Ai 1 A (4.6.9), provided 1 > A (4.6.10) 86

87 Equation (4.6.10) guarantees K =1 which is better than K >1 because in the former, at least the error only grows in accordance with the solution instead of blowing up in the later situation. However, Eq. (4.6.10) is valid only when A 11 A11 1 sin 1 sin 1 (4.6.11) Finally, Eq. (4.6.11) is true whenever ct x (4.6.1) Equation (4.6.1) is the required condition for the explicit scheme Eq. (4.6.5), to the wave Eq. (4.6.3), to be stable. 1 87

88 CFL ( Courant Friedrichs Lewy ) Condition It was stated during the classification of the PDE that, hyperbolic equations have a domain of dependence. the To illustrate this, consider the D Alembert s solution of the wave Eq. (4.6.3)-(4.6.4), with x (-,), given by xct uxt (, ) 1 f ( xct) f ( xct) 1 g( vdv ) c xct (4.6.13) Equation (4.6.13) demonstrates that the solution of the wave equation at any point (x, t) depends only in the triangular region bounded by the left running characteristic, the right running characteristic and the x-axis. 88

89 Therefore, any numerical scheme which is developed to solve Eq. (4.6.3) should also use all the data from this region. The condition which guarantees such a criteria is called CFL condition. Since, Eq. (4.6.13) uses f and g in the region (x i -at n,x i+ at n )onthe x-axis to compute the solution at any given point (x i, t n ), therefore, according to CFL condition (the numerical domain of dependence should bound the analytical domain of dependence) that is c t xi n xi ctn xi n x xi cn t x c t x 1(4.6.14) 89

90 Condition Eq. (4.6.14) is equivalent to the stability condition σ 1 for the explicit scheme Eq. (4.6.5). Further, if c=1 and the step lengths t= x, then from Eq. (4.6.8), the scheme Eq. (4.6.5) is fourth order accurate. Therefore, if the step lengths are fixed such that σ = 1, then the explicit scheme Eq. (4.6.5) gives the best solution (in this particular case, that is, for c=1 and t= x, the entire right hand side of Eq. (4.6.8) becomes zero and Eq. (4.6.5) produces exact solution). 90

91 Numerical Illustration Compute the solution of the following wave equation using the explicit scheme Eq. (4.6.5) for 10 time steps. u u,0 x1, t 0 t x u ux (,0) sin( x), ( x,0) 0, 0 x1 t u(0, t) u(1, t) 0, t 0 The exact solution of the problem Eq. (4.6.15) is (4.6.15) sin(x) cos(t) (4.6.16) 91

92 Fixing x =0.1and t = (so that σ = 0.75), scheme Eq. (4.6.5), after 10 time steps (that is at time t =.75), produces the a numerical solution which has been compared with the analytical solution in the Table 4.6.1, The percentage error is 0.38%. The numerical solution in the Table is generated using the exact solution at the time step t. In the computations if t is changed to 0.1 (σ = 1), as discussed earlier, the explicit scheme has produced the exact solution (except for some machine error at ). 9

93 x=0.1 x=0. x=0.3 x=0.4 x=0.5 Exact Sol Num. Sol Table Comparison of exact and numerical solutions with σ = 0.75 In the above computations, if t is fixed such that σ =1.5then after 30 time steps the explicit scheme produces a numerical scheme with error more than 500% because with σ = 1.5, the scheme violates both stability and CFL conditions. 93

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