Lecture 4.5 Schemes for Parabolic Type Equations

Transcription

1 Lecture 4.5 Schemes for Parabolic Type Equations 1

2 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, ( a, b ), t 0 (4.5.1) Assume that the initial conditions, the distribution of u at t = 0 and the boundary conditions, u at = a and b are given.

3 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 eplicit method. An eplicit scheme uses a stencil in which only one unknown is written in terms of the remaining known values at other stencil points. The FTCS approimation of Eq. (4.5.1) is 1 n 1 n 1 n n n ui ui k u i 1 ui ui 1 t n 1 n n n n t ui ui r ui 1 ui ui 1, r k u ru (1 r) u ru i 1,,, n 1, n 0,1, n 1 n n n i i 1 i i 1 (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

4 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 -1. 4

5 Taylor series epansion 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 approimation of Eq. (4.5.1) gives 1 n n 1 1 n n n ui ui k u i 1 ui ui 1 t n n n n 1 t rui 1 (1 r) ui rui 1 ui, r k, i 1,,, n 1, n 1,, (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. 5

6 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. 6

7 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 n 1 n 1 n 1 n t rui 1 (1 r) ui rui 1 ui, r k, i 1,,, n 1, 0,1, n (4.5.4) 7

8 Weighted average scheme Weighted average of Eqs. (4.5.1) and (4.5.3), gives 1 n 1 n k n n n n 1 n 1 n 1 ui ui (1 ) u i 1 ui ui 1 ui 1 ui ui 1 t r u (1 r) u r u (1 ) r u (1 (1 ) r) u (1 ) r u n 1 n 1 n 1 n n n i 1 i i 1 i 1 i i 1 t r k, 0 1, i 1,,, n, 0,1, n (4.5.4) For θ equals to 0 and 1, Eq. (4.5.4) gives FTCS and BTCS schemes, respectively. The Taylor series epansion of (4.5.4) gives u u 1 u t u k u k t u k t t t 1 t n 3 n 4 n 4 n i i i i t 3 4 (4.5.4) 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. 8 1

9 Numerical Illustration u Consider, (0,1), t with initial conditions u(,0) = sin t 0 and boundary conditions zero at = 0 and 1. Use step sizes 0. and 0.01 in 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. For the step size 0., we have u r k t 1* *0. With r = 0.3, the FTCS, BTCS and Crank-Nicolson schemes are given by 0.3 9

10 u 0.3u 0.4u 0.3u n 1 n n n i i 1 i i 1 0.3u 1.6u 0.3u u n 1 n 1 n 1 n i 1 i i 1 i 0.15u 1.3u 0.15u 0.15u 0.7u 0.15u n 1 n 1 n 1 n n n i 1 i i 1 i 1 i i 1 for i 1,,3, 4, n 0,1,,,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

11 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 eact solution. 11

12 Two-dimensional problems Consider the unsteady diffusion over a flat plate governed by the differential equation u u u k, (, y) ( a, b) X ( c, d), t 0 t y (4.5.7) Assume that the initial and boundary conditions on u are known. Eplicit scheme Approimating the time derivative with forward difference and space derivatives with central differences gives a scheme u u k u u u u u u t y n 1 n n n n n n n i, j i, j i 1, j i, j i 1, j i, j 1 i, j i, j 1 u u r u u u r u u u n 1 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 n 1 n n n n n ui, j r ui, j 1 ru 1 i 1, j (1 r1 r ) u i, j ru 1 i 1, j r ui, j 1 t r k, r k t y 1 i 1,,, n 1, j 1,,, n 1, n 0,1, y (4.5.8) 1

13 Here, y is the step length in y direction. Taylor series epansion of Eq. (4.5.8) shows that the eplicit scheme is first order accurate in time and second order in space (both in and y directions). Weighted average or Crank-Nicolson type of approimation 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 epensive computationally, therefore, alternatively ADI method can be developed as follows: n n n n n n n n i, j i, j i 1, j i, j i 1, j i, j 1 i, j i, j 1 k u u u u u u u u t / y (4.5.9) n n n n n 1 n 1 n 1 n 1 i, j i, j i 1, j i, j i 1, j i, j 1 i, j i, j 1 k u u u u u u u u t / y 13

14 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 -1 + n y -1) tridiagonal systems in 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 -1) * (n y -1)) appears in the weighted average or Crank-Nicolson schemes for two dimensional problems. 14

15 Convergence Hear, 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 La equivalence theorem, Consistency and stability are necessary and sufficient conditions for convergence of the finite difference solutions of any time dependent problem. 15

16 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. 16

17 Numerical Illustration Consider the Taylor series epansion Eq. (4.5.5) of the Weighted average scheme Eq. (4.5.4) r u (1 r) u r u (1 ) r u (1 (1 ) r) u (1 ) r u given by n 1 n 1 n 1 n n n i 1 i i 1 i 1 i i 1 u u 1 u t u k u k t u k t t t 1 t n 3 n 4 n 4 n i i i i t 3 4 Taking step lengths and t tending to zero, the series epansion converges to the governing equation k, therefore, the weighted average scheme is consistent. u t u 17

18 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 18

19 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 =

20 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: 0

21 n Let (u) = 0 be a linear difference scheme and is its numerical n n solution, U is the eact solution and is its error at n th i E i time level at the nodal point i then we have n n n u (4.5.10) i Ui Ei and n n n n ( ui ) = ( U E ) = ( ) + ( E n ) = ( E n ) = 0 (4.5.11) i i 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 1

22 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 epressible it 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 where j i j u K e K e K e (4.5.1) N is the number of points in the discretization, (N=L/ ) I n K j 1 is the amplitude of the j th harmonic, k j is the wave number n N n Ik N n Ik i N n Ii i j j j j N j N j N is the phase angle given by k j.

23 Note: k j varies from N to N instead of - to, because the maimum and minimum resolvable wavelengths (λ) are only L and, respectively and the maimum 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 n 1 Ki G (4.5.13) K n Any scheme (linear) is said to be stable if Otherwise is said to be unstable. i G < 1 (4.5.1) 3

24 Numerical Eample: Discuss the stability criteria of the weighted average scheme r u (1 r) u r u (1 ) r u (1 (1 ) r) u (1 ) r u n 1 n 1 n 1 n n n i 1 i i 1 i 1 i i 1 (4.5.15) n n Ii Substituting u K e in Eq. (4.5.15) and simplifying for i amplification factor G, gives G n 1 I I K (1 ) re (1 (1 ) r) (1 ) re 1 4(1 ) r sin n I I K re (1 r) re 1 4 r sin (4.5.16) For θ = 0, that is, for FTCS scheme, G <1 implies which is true whenever the value of r < 1/. 1 4r sin 1 4

25 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 for r = 0.45, FTCS scheme is able to produce accurate solutions 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

26 Eample: Analyze the stability of the scheme t u u r u u u r u u u, r k, r k n 1 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 1 t y n Ii Ii n n y Substituting either Ke or Ke for u ij, depending on variation of u in or y directions we get n 1 K i i i y G 1 r1 e e r e e n K 1 r cos 1 r cos r1sin 4rsin (4.5.17) Then the condition G <1 is satisfied whenever r 1 +r < ½. If r 1 = r then condition for the stability of the two dimensional diffusion problem is r < ¼ which is even stronger than the 6 condition of the corresponding 1D case. y y i y

27 Eercise Problems Discuss the consistency and stability criteria for the following schemes: 1. Crank-Nicolson (CN) scheme for the two-dimensional diffusion problem u u r u u 3u u u n 1 n n n n n n i, j i, j 1 i, j i 1, j i, j i 1, j i, j r u u 3u u u n n n n n i, j 1 i, j 1 i, j i, j 1 i, j 3. For the scheme obtained by discretizing the time derivative with forward difference approimation and space derivatives with second order central difference approimations of the equation, u a u where a, a y and μ are a u u u y t y y 7 constants.

28 Summary of Lecture 4.5 Various finite difference approimations of parabolic equations like unsteady diffusion equations are introduced in this lecture. END OF LECTURE 4.5 8