Fusion Higher -Order Parallel Splitting Methods for. Parabolic Partial Differential Equations

Transcription

1 International Mathematical Forum, Vol. 7, 0, no. 3, Fusion Higher -Order Parallel Splitting Methods for Parabolic Partial Differential Equations M. A. Rehman Department of Mathematics, University of Management and Technology Lahore, Pakistan S. A. Mardan Department of Mathematics, University of Management and Technology Department of Sciences and Humanities, National Univ. of Computer and Emerging Sciences Lahore Campus, Pakistan M. S. A. Taj Department of Mathematics, COMSATS Institute of information Technology Lahore, Pakistan A. A. Bhatti Department of Sciences and Humanities National Univ. of Computer and Emerging Sciences Lahore Campus, Pakistan Abstract: A family of numerical methods, based upon a rational approximation to the matrix exponential function, was developed for solving parabolic partial

2 568 M. A. Rehman, S. A. Mardan, M. S. A. Taj and A. A. Bhatti differential equations. These methods were partially sixth-order precise in space and time, due to combination of sixth-order finite approximations and fifth-order pde s approximations. These methods do not involve the use of complex computation. In these methods second-order spatial derivates were approximated by sixth-order finite difference approximations. Parallel algorithms were developed and tested on the one, two and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions and time dependent boundary conditions. It was observed that the results obtained through these methods were highly accurate and can be easily coded on serial or parallel computers. Keywords: Heat equation, Fifth order numerical methods, Higher order pde s approximations, Method of lines, Parallel algorithm Introduction Main idea behind the finite-difference methods for obtaining the approximate solution of a given PDE is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points. The most usual way to generate these approximations is through the use of Taylor series approximation. The basis of analysis of the finite-difference equations considered here is the modified equivalent PDE approach. It is worth noting that from the truncation error of the modified equivalent equation, it is possible to eliminate the dominant error terms associated with the finite-difference equations that contain weights, thus leading to more accurate techniques. Higher dimensional parabolic initial/boundaryvalue problems are studied by several authors. Twizell et al. develop families of third and fourth-order methods, which do not require complex arithmetic [5], [6]. M. Dehghan starts working on the concept of parallel splitting method and apply similar techniques for the solution of parabolic and hyperbolic PDEs [3]. Borovykh, Day. W. A and S. Wang also discussed the solution of parabolic PDE but with non-local boundary conditions. One ways of solving parabolic partial differential equations is by the use of method of lines (MOL), which transform parabolic partial differential equations into a system of ordinary differential equations which can be written in matrix-vector form as () Where is a square matrix, results from non-homogeneous boundary conditions and is the solution vector at time. Then solution satisfies the recurrence relation exp exp; 0,,, () when boundary conditions are homogeneous, then recurrence relation takes the form exp; 0,,, (3)

3 Fusion higher-order parallel splitting methods 569 Derivation of the method A typical problem in applied mathematics is the one-dimensional homogeneous heat equation with time dependent boundary conditions. This initial/boundary value problem is given by, 0, 0 (4) and the non-local boundary conditions 0,, 0 (5),, 0 (6) subject to the given initial condition,0, 0x (7) where is a given continuous function of. There will exist discontinuities between the initial conditions and the boundary conditions if 0 0 and/or 0. Dividing the interval 0, into subintervals each of width, so that, and the time variable into time steps each of length gives a rectangular mesh of points with co-ordinates,, 0,,,,, and 0,,, covering the region 0 and its boundary consisting of the lines 0, and 0. The space derivative in (4) may be approximated to the sixth-order accuracy at some general point, of the mesh by the expression, ux3h,t7uxh,t70uxh,t490ux,t 70uxh,t7uxh,t ux3h,t, Oh (8) However, equation (8) is valid only for x,t, with 3,4,, and 0,,,3,4,. To attain the accuracy at the same points at,,,,, and, special formulas must be developed which approximate, not only to sixth-order but also with dominant error term can be clearly shown that the desired approximations to, are,. It, 7uxh,tux,t738uxh,t 359uxh,t 300 ux3h,t88 ux4h,t34 ux5h,t, 80, ux6h,t9ux7h,t, Oh (9) 9uxh,t98uxh,t3 ux,t 8,5,66ux3h,t7ux4h,t 8ux5h,tux6h,t, Oh (0) 9uxh,t98uxh,t3 ux,t 8 uxh,t5uxh,t66 ux3h,t

4 570 M. A. Rehman, S. A. Mardan, M. S. A. Taj and A. A. Bhatti 7 ux4h,t8 ux5h,tux6h,t, Oh (), 80 7uxh,t ux,t738 uxh,t 359uxh,t300 ux3h,t88 ux4h,t34 ux5h,t 83ux6h,t9ux7h,t, Oh () at the mesh points,,,,, and, respectively. Appling (4) with (8)-() to the mesh points of the grid at time level produces a system of ODE s of the form, 0 (3) With initial distribution 0 (4) in which,,,,,,,, denotes transpose A = h O O O O O O O (4a) with 4,3,0,,,4,9,8 and the solution of system {(3)-(4)} satisfy the recurrence relation (). To approximate the matrix exponential function in () fifth-order pde s approximation, for a real scalar, given by (5) where (6)! and, 0,,,3,4 (7)! The integral term in () is approximated as exp (8)

5 Fusion higher-order parallel splitting methods 57 Where and,,3,4,5 are matrices. We have exp,,,3,4,5 (9) with exp,,,3,4,5 (0) Taking,,,,. Using in (5) and taking exp, we have () () and () (3) (4) (5) (6) (7) Algorithm Assuming that,,,, 0 are real distinct zeros of, the denominator of, then (8) exp (9) where,,3,4,5 and

6 57 M. A. Rehman, S. A. Mardan, M. S. A. Taj and A. A. Bhatti Hence equation (4) becomes where Hence,,,3,4,5 where,,,3,4,5 are the solutions of the systems

7 Fusion higher-order parallel splitting methods Extension to two-space dimension The above method cannot be extended to two/three-space dimensions with timedependent boundary conditions. Consider the two-dimensional heat equation with constant coefficients,,,,,,, 0,, 0 (30) subject to the given initial condition,,0,, 0xX (3) where, is some continuous function of and and the boundary conditions 0,,,,0, 0 (3),0,,,0, 0 (33) Discretizing 0, as in the one-dimensional case using equal space steps and replacing the space derivatives in the PDE (30) by the appropriate fifth-order approximations and applying to all -interior mesh points at time level 0,,,3, gives a system of first-order differential equations which may be written in matrix-vector form as, 0 (34) With initial distribution 0 (35) In which,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, denoting the transpose, the matrix is the sum of two square matrices and of order, given by B = 80h 80h A 80 h A O 80h A N N Where matrix A is given by (4a) and

8 574 M. A. Rehman, S. A. Mardan, M. S. A. Taj and A. A. Bhatti B = 80 h I 738I 359I 300I 88I 34I 83I 9I 98I 33I 8I 5I 66I 7I 8I I 7I 70I 490I 70I 7I I I 7I 70I 490I 70I 7I I I 7I 70I 490I 70I 7I I O O O O O O O I 7I 70I 490I 70I 7I I I 7I 70I 490I 70I 7I I 8I 7I 66I 5I 8I 3I 98I 9I 83I 34I 88I 300I 359I 738I I Where is identity matrix of order and the solution of system (34), (35) satisfies the recurrence relation (3) Clearly and commute. Development of Algorithm Since, replacing by in equation (3) gives exp exp,,,3, (36) to the fifth order. Using (9) in (36) gives (37) Let,,,3,4,5 (38) Then the linear systems,,3,4,5 (39) can be solved for vectors,,3,4,5 of order of five different process simultaneously. Assuming that (40) Equation (37) can be written as or (4) in which,,3,4,5, the solution of linear systems,,,3,4,5 (4) can be computed on five different processors simultaneously.

9 Fusion higher-order parallel splitting methods 575 Tabular form of algorithm Steps = input Processors,,3,4,5,,,, = compute,, 3=Decompose, 4= Solve 5= Calculate 6=Solve 7= Calculate Go to Step 4 for next time step 3 Extension to three- space dimension Consider the three-dimensional heat equation with constant coefficients,,,,,,,,,,,,, 0,,, 0 (43) subject to the given initial condition,,,0,,, 0x,,X (44) where,, is some continuous function of, and and the boundary conditions 0,,,,,,0, 0,X,0 (45),0,,,,,0, 0x,X,0 (46),,0,,,,0, 0x,X,0 (47) Discretizing 0,, as in the one-dimensional case and replacing the space derivatives in the PDE (43) by the appropriate sixth-order approximations and applying to all -interior mesh points at time level 0,,,3, gives a system of first-order differential equations which may be written in matrix-vector form as, 0 (48) With initial distribution 0 (49) In which,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, denoting the transpose, the matrix is the sum of three square matrices, and of order, given by

10 576 M. A. Rehman, S. A. Mardan, M. S. A. Taj and A. A. Bhatti C = 80h 80h A 80h A O 80h A N 3 3 N Where matrix A is given by (4a) and is a block diagonal matrix with diagonal blocks. 80h B C = 80h 80h B O 80h B N 3 N 3 Where is a block-diagonal matrix with diagonal blocks and C 3 = 8 0 h I I I I 8 8 I 3 4 I 8 3 I 9 I I I I I I I I I 7 I 7 0 I I 7 0 I 7 I I I 7 I 7 0 I I 7 0 I 7 I I I 7 I 7 0 I I 7 0 I 7 I I O O O O O O O I 7 I 7 0 I I 7 0 I 7 I I I I I I I I I 8 I 7 I 6 6 I 5 I 8 I 3 I 9 8 I 9 I 8 3 I 3 4 I 8 8 I I I I I Where I is the identity matrix of order. Development of Algorithm Since, replacing by in equation (3) gives exp exp exp,,,3, (50) So exp (5) and Let Then solving the linear systems (5),,,3,4,5 (53)

11 Fusion higher-order parallel splitting methods 577,,,3,4,5 (54) for,,3,4,5 (55) where (56) Let,,,3,4,5 (57) then the solution of the linear system,,,3,4,5 (58) Gives (59) Then (60) In which,,3,4,5 are the solution of the linear systems,,,3,4,5 (6) The algorithm used here is given in tabular form as Tabular form of algorithm Steps Processors,,3,4,5 = input,,,,, = compute,,, 3=Decompose,, 4= Solve 5= Calculate 6=Solve 7= Calculate 8= Solve 9= Calculate Go to Step 4 for next time step 4 Numerical experiments Using above given algorithms these problem are solved for different values of 9 and. In these experiments the method behaves smoothly over the whole interval 0 and no oscillations are observed. Now the developed numerical methods were applied to different problems from the literature by taking

12 578 M. A. Rehman, S. A. Mardan, M. S. A. Taj and A. A. Bhatti , ,.3375,.99400, , , 5.404, ,.00, Example (One-dimensional Problem) Consider the Second order PDE problem () with, 0, exp and sin. This problem which has analytical solution, exp 4 sin has a time-dependent boundary condition but does not have discontinuities between the initial conditions and boundary conditions. The maximum value of the analytical solution of this problem at time.0 occurs for.0 and is approximately Example (One-dimensional Problem) Consider the Second order PDE problem with,, 0 and. This problem which has analytical solution, 6 3 exp sin has discontinuities between the initial conditions and boundary conditions at 0 and. The maximum value of the analytical solution of this problem at.0 occurs for 0 and is Example 3 (One-dimensional Problem) Consider the Second order PDE problem with, 0, 0 and. This problem which has analytical solution, sin exp 4 has discontinuities between the initial conditions and boundary conditions at 0 and. The maximum value of the analytical solution of this problem at.0 is e-00 Example 4 (Two-dimensional Problem) Consider the two space dimensional heat equation with constant coefficients with and,sin the model problem has theoretical solution,,sin sin exp 4

13 Fusion higher-order parallel splitting methods 579 The maximum value of the analytic solution occurs at.0 for,, and is approximately Example 5 (Three-dimensional Problem) Consider the three space dimensional heat equation with constant coefficients with and,,sin sin sin the model problem has theoretical solution,,,sin sin sin exp3 4 The maximum value of the analytic solution occurs at 0. for,,,, and is approximately Conclusion It is observed that the results obtained using new scheme are highly accurate and the method developed is fifth-order accurate in space and time. This technique can be coded easily on serial or parallel computers. It is worth mentioning that the methods using real arithmetic and multiprocessor architecture especially in multidimensional problems will save remarkable CPU time rather than the complex arithmetic based methods. Table. Maximum absolute error for Example e e e e e e e e e e-00.65e e e e e e-00 Table. Maximum absolute error for Example e e e e e e e e e e-00.85e-00.49e e e e e-00 Table 3. Maximum absolute error for Example e e e e e e e e e e e e e e e e-009

14 580 M. A. Rehman, S. A. Mardan, M. S. A. Taj and A. A. Bhatti Table 4. Maximum absolute error for Example e e e e e e e e e e e e e e e e-009 Table 5. Maximum absolute error for Example e e e e e e e e e e e e e e e e-007 References [] Borovykh, Stability in the numerical solution of the heat equation with nonlocal boundary conditions, Appl. Numer. Math, 4 (00), 7-7. [] W. A. Day, Extension of a property of the heat equation to linear thermo elasticity and other theories, Quart. Appl. Math. 40 (98), [3] Deghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math, 5 (98), [4] Liu, Numerical solution of the heat equation with nonlocal boundary conditions, J. Comp. Appl. Math, 0 (999), 5-7. [5] M. S. A. Taj and E. H. Twizell, A family of third-order parallel splitting methods for parabolic partial differential equations, Intern. J. Comput. Math. 67 (998), [6] M. S. A. Taj and E. H. Twizell, A family of fourth-order parallel splitting methods for parabolic partial differential equations, Intern. J. Comput. Math. 4 (998), [7] S. Wang, and Y. Lin, A numerical method for the diffusion equation with nonlocal boundary specifications". Inter. J. Engrg. Sci. 8 (990), Received: January, 0