Numerical Solution to Parabolic PDE Using Implicit Finite Difference Approach

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1 Numerical Solution to arabolic DE Using Implicit Finite Difference Approac Jon Amoa-Mensa, Francis Oene Boateng, Kwame Bonsu Department of Matematics and Statistics, Sunyani Tecnical University, Sunyani, Gana Department of Interdisciplinary Studies, University of Education, Winneba,. O. Box, Kumasi, Gana Abstract Tis paper examines an implicit Finite Difference approac for solving te parabolic partial differential equation (DE) in one dimension. We consider te Cran Nicolson sceme wic offers a better truncation error for bot time and spatial dimensions as compared wit te explicit Finite Difference metod. In addition te sceme is consistent and unconditionally stable. One downside of implicit metods is te relatively ig computational cost involved in te solution process, owever tis is compensated by te ig level of accuracy of te approximate solution and efficiency of te numerical sceme. A pysical problem modelled by te eat equation wit Neumann boundary condition is solved using te Cran Nicolson sceme. Comparing te numerical solution wit te analytical solution, we observe tat te relative error increases sarply at te rigt boundary, owever it diminises as te spatial step size approaces zero. Keywords: artial Differential Equation, Implicit Finite Difference, Cran Nicolson Sceme. Introduction Myriads of pysical penomenon lie diffusion, wave propagation, laser beam models, financial models, etc., are modelled by partial differential equations. Most often te equations under consideration are so complicated tat finding teir solutions analytically (e.g. by Laplace and Fourier transform metods, or in te form of a power series) is eiter impossible or impracticable (Süli, ). Tis may be due to several reasons including nonlinearity of problems, variable coefficients, difficulty in evaluating integrals analytically, approximation based on truncation of infinite series required, inappropriate solution space, etc. And, even problems tat could be solved analytically, we depend on numerical procedure to plot te solution. Consequently, numerical metods are employed as alternative approac for te approximation of te unnown analytical solution (Bruaset, and Tveito, 6). Te main idea of any numerical metod for a differential equation is to discretize te given continuous problem wit infinitely many degrees of freedom to obtain a discrete problem of system of equations wit only finitely many unnowns tat may be solved using numerical algoritms (Claes, 9). Altoug tere are several different numerical metods available for solving te parabolic equations, we focus on finite difference scemes. Tis is motivated by te fact tat tey are very simple to understand and tey are easy to generalize to more complex boundary value problems, and also are easily implemented on a computer. Te finite difference metod comes in two forms namely; te explicit metods and te implicit metods. Explicit metods generally are consistent, owever teir stability is restricted (LeVeque, ). On te oter and te implicit metods are consistent as well as unconditionally stable, owever tey are computationally costly compared to te explicit metods (Douglas and Kim, ). Tis is compensated by te ig level of accuracy of approximate solution and efficiency of te numerical sceme. In tis paper we use te Cran Nicolson metod wic is an implicit finite difference metod to solve a pysical problem modelled by te parabolic equation. We coose tis metod over te explicit metod because of its attractive properties mentioned.. Finite Difference Metod In tis section, we discuss te concept of te finite difference metod by providing useful terminologies, definitions and teorems. Te differencing metod solves a DE numerically by setting up a regular grid in space and time to compute approximate solutions at space or time points of tis grid (Strauss, 8). Te metod consist of approximating te differential operator by replacing te derivatives in te equation using differential 4

2 quotients. Tis process is commonly termed as discretization. Te error between te numerical solution and te exact solution is determine by te error tat is committed by going from a differential operator to a difference operator. Tis error is called te discretization error or truncation error. Definition.. Suppose te function u is C continuous in te neigbourood of x. For any > we ave u( x ) u( x) u( x) u ( x ) () were is a number between and : x ] x, x [. Truncating () at te first order gives u( x ) u( x) u( x) O( ) () were te term O ( ) indicates tat error of te approximation is proportional to We refer to equation () as te forward difference approximation of u. We deduce from equation () tat tere exists a constant C suc tat, for > sufficiently small, we ave. were u( x ) u( x) u( x) C, C sup [ x, x ] u( ) for, given. Te approximation of u at te point x is said to be consistent at te first order. Te order of a first derivative can be defined in a more general sense as follows; Definition.. Te approximation of te derivative u at point x is of order p (p > ) if tere exists a constant C p >, independent of, suc tat te error between te derivative and its approximation is bounded by C, i.e., p is exactly O ( ). Similarly, te first order bacward difference approximation of u at point x can be defined as u x u x u x O ( ) ( ) ( ) ( ) (3) In order to improve te accuracy of te approximation of u we define a consistent approximation, called te central difference approximation, also referred to as symmetric quotient, by combining te forward difference and te bacward difference (Lapidus, and inder,999). Tat is, taing te points x and x into consideration. Definition.3. Suppose tat te function uc 3 ( ) were ] x, x [, ten we write, 3 u( x ) u( x) u( x) u ( x) u ( ) (6) 6 3 u( x ) u( x) u( x) u ( x) u ( ) () 6 were ] x, x [ and ] x, x[. Subtracting () from (6) to we obtain, 3 u( x ) u( x ) u( x) u ( ) (8) 6 were ] x, x [. It follows from te definition tat, for every ], [, we ave a bound on te approximation error as: 5

3 u( x ) u( x ) u x ( ) C, C sup [ x, x] u( ) 6 If u is C continuous, ten te approximation is consistent at te order one only. Te order of te approximation is related to te regularity of te function u (Alan, ). Te approximation of te second derivative and its associated truncation error is given by te following lemma. Lemma.. Suppose u is a C 4 continuous function on an interval[ x, x ],. Ten, tere exists a constant C > suc tat for every ], [, we ave u( x ) u( x) u( x ) u( x) C. (9) Te differential quotient ( u( x ) u( x) u( x )) is consistent second-order approximation of te second derivative u of u at point x. roof. We begin by using te Taylor expansion up to te fourt order to obtain te following: u x u x u x u x u x (4) ( ) ( ) ( ) ( ) ( ) u ( ) () u x u x u x u x u u 6 4 were ] x, x [ and ] x, x[. By te intermediate value teorem, we can write: 3 4 (4) ( ) ( ) ( ) ( ) ( ) ( ) () u( x ) u( x) u( x ) u x u for ] x, x [. Hence, from equation (9) we deduce te constant (4) u ( ) C sup. [ x, x] (4) ( ) ( ), Remar.. Te error estimates (9) depends on te regularity of te function u. If u is C 3 continuous, ten te error is of order only. 3. arabolic roblem in -dimension arabolic DE describes time dependent pysical penomenon and is mostly used in modeling diffusion problems suc as eat transfer (Morton and Mayer, 5). In tis section, we consider te one-dimensional omogenous eat equation posed in te bounded domain [, L] : u u, for ( xt, ) t t u( x,) u( x), for x, () BC : u(, t), u( L, t), for, were is a given scalar value and u C (, ). Te boundary condition (BC) specified in () is of te Diriclet boundary condition. Te boundary condition can be replaced by Neumann boundary condition given as, u u (, t), ( L, t),, x x 3.. Analytic Solution Te general analytic solution of te eat equation () wit omogeneous Diriclet boundary condition is as follows (Kersale, 3): 6

4 n nx t L n n L u( x, t) b sin e were L n bn u ( x)sin dx, n,,3, L L Tis solution can be obtained by using te metod of separation of variables and Fourier transform. Wen te boundary conditions are of te Neumann ind ten te solution is written as: were n t L nx n n L u( x, t) a a e cos L n an u ( x)cos dx, n,,, L L 3.. Implicit Finite Difference Sceme To begin wit, we introduce equi-distributed grid points ( x ) given by i i N xi i, wit L ( N ) to discretize te domain, in space. Furtermore, we consider time step, to discretize te time domain, ence defining a regular grid, t,. We denote by U( x, i t ) te value of te numerical solution at point ( x, i t ) and u( x, t ) te exact solution of te equation (). Te initial data u is discretized by: Ui u ( xi) for i {,, n }. Te boundary conditions are also andled in similar vein. Diriclet boundary conditions is represented in a discrete form in a straigt forward manner as: U, UN. In te case of Neumann boundary conditions, in order to improve te accuracy of te solution we apply te central difference approximation to discretize te derivatives involved. Tis is given by N N U U, U U, Using te finite difference approximations for te derivative in te eat equation we obtain te following: u u( x, t ) u( x, t) ( x, t) O( ) (3) t u u( x, t) u( x, t) u( x, t) ( x, t) O( ) (4) x Considering equations (3) and (4), te finite difference scemes can be written in a more general form as a weigted average of te explicit and implicit scemes, producing te teta sceme, also written as te -sceme : U i Ui Ui Ui ui Ui Ui Ui ( ) (5) We simplify equation (5) furter as: Ui Ui U i Ui U i ( ) U i Ui u i (6) were denotes a positive parameter, [,] and Te forward and bacward Euler metods, and te Cran Nicolson metod are considered to be special cases of te -sceme defined by (6), wit for te forward Euler, for bacward Euler, and for te Cran Nicolson metod. Te -sceme is an implicit sceme for.

5 Te difference equation of te -sceme (6) can be written in a vector form as: were I is a unit matrix, and I A U I ( ) A U () T,,, N U U U U... A Tus, to find U, we need to solve a tridiagonal linear system, wic can be done by employing Tomas algoritm. Lemma 3.. Let T be an N N tridiagonal matrix of te form: b c.. a b c. T a b c.. a b Te eigenvalues and te corresponding eigenvectors of T are: b a sin c N a sin ac cos, v c N, N N a N sin c N,, N (Kicaid and Ceney, ) Lemma 3.. Consider te -sceme (4), for [,]. (i) If, te -sceme is unconditionally stable in L - norm (and convergent) (ii) If, te -sceme is stable in L - norm (and convergent of tis norm) under te condition:. ( ) roof. To establis te stability of te -sceme, we represent te sceme () in a matrix form as U U (8) and we determine weter te magnitude of any eigenvalue of coefficient matrix exceeds. From equation (), we can write, It follows from (9) tat, U I A I ( ) A U (9) I A I ( ) A 8

6 Since te eigenvalues of I, are easily found be ( ) A and I A ( ) are te same, te eigenvalues appearing in te matrix were are te eigenvalues of A. Te values of can directly be deduced from lemma 3.. as, cos 4sin, N N,, N () For stability of te sceme, it is necessary tat, We denote ( ) N,,, N () () Substituting equations () and () into () gives, 4 ( ) sin 4 sin. (3) Since, by assumption and by definition, ten te rigt and side of (3) is positive, and terefore te inequality can be written as ( 4 sin ) 4 ( ) sin 4 sin (4) Te rigt part of te inequality is automatically satisfied for all. Te left part is satisfied wen Te strongest restriction on (and ence on te step size in time) occurs wen 4 ( ) sin (5) sin value, tat is. In tat case, (5) yields (6) assumes its largest Considering (6), we note tat te -sceme is unconditionally stable if, for any arbitrary. If, ten te -sceme is stable provided tat, 3... Cran Nicolson Sceme We consider Cran Nicolson metod for solving te eat equation in tis paper due to te fact tat te metod as second order accuracy in bot time and spatial dimensions, and is unconditionally stable (Arivis, et. al., 6). Tis means tat te stability of te sceme is not affected by any step size cosen for te time dimension as seen in lemma 3.. Te metod as mentioned earlier can be extracted from te -sceme by setting. We can write te sceme as follows: U i Ui Ui U i ui Ui U i Ui Rearranging () gives, () i i i i Ui U U i U Ui U Ui U (8) 9

7 were Now, before we generate te system of linear equations out of (8), we recall te discrete form of te boundary conditions discussed earlier. Let us consider for instance te discrete omogeneous Neumann boundary condition given by, N N U U U U For i,, N, we generate a system of linear equations out of equation (8), taing into consideration te boundary conditions. We obtain te following system: U U U U U U U U U U U U U U U U U U U U U U U U U U 3 3 Te above system can be written in a matrix form as U U U U U ( ) U U ( U U ) ( ) U U ( U U U ) ( ) U U ( U 3 U U ) (9) ( ) U U9 ( U U9 ) Eac time step, requires tat we solve te tridiagonal system (9). Tis maes it computationally costly compared to te explicit metods. However, for one dimensional eat problem, tere is not muc difference between te explicit metod and te implicit metod in terms of computational cost. 4. Test Case Consider te equation describing one-dimensional, single-pase, sligtly compressible flow in a producing petroleum reservoir wic is given, for x 5 and t, by;, px, t C p x t K t x, (3) te porous medium and te reservoir are omogeneous, tat te liquid is ideal, and tat gravitational effects are negligible. Te distance is represented by x, time by t, p te pressure, te dimensionless constant porosity of te medium, te viscosity te permeability of te medium and C te compressibility (in {pounds per square inc} - ). Assuming tat C K.4 days/ft, and tat te following conditions old: p x,.5, x 5,, 5, p t p t, t. Kx x 8

8 Determine te pressure p at t = 5, using te Cran Nicolson metod wit t.5 and x 5 (Burden et. al., 98). Te analytical solution for te eat problem described above is given by n.4 5 t 5 nxn p( xn, t).5 sin ne cos n n 5. Te pressure p, at time t = 5 given at te spatial step, x 5 is provided as follows x px (,5) x x x x x x x x x x x 4.. Numerical Solution We solve te eat problem (3) numerically using te Cran-Nicolson metod wit t.5 and x 5. We represent te approximate solution of te eat problem (3) at te next time step by ( x, t ), wic for simplicity can be written as. Expressing (3) by te Cran Nicolson sceme (8), we i obtain i i i i i i i i i (3) To incorporate te Neumann boundary condition,, 5, t t, Kx x we discretize te boundary data using central difference to obtain, i i. Te spatial domain is discretized using x = i. At te left boundary i = and so x =. Te boundary reduces to. We andle te rigt boundary in similar manner by setting i =, wic implies x = 5. Te boundary can ten be written as 9 Now, we generate a system of equations wic will enable us solve for te interior nodes. For i = in equation (3) we ave, But and 8

9 For i For i = 3 3 For i = 3, 4., we get a system of equations wic can be written in a matrix form as ( ) ( ) ( ) ( ) ( ) ( 3 ) ( ) 9 ( 9 ).5 and.5. Following from te eat problem (3), we let Te initial conditions, x,.5 can also be written in a discrete form as follows: i.5.5 (3).5 Substituting te value of and te initial condition into (3), we obtain a tridiagonal system: (33) Hence te approximate solution as a result of solving te tridiagonal system (33), for =,,,... 9, is given as follows: 8

10 xi i x (,5) i Te solution of bot te analytical approac and Cran Nicolson sceme for spatial step, = 5 and = are compared respectively in figure. Also te values of te relative error of bot solutions are plotted in figure. Figure : Comparison of exact solution and approximate solution for = 5 and =. Figure : Relative error of te approximate solution for = 5 and =. Clearly, we see from figure tat, te Cran Nicolson metod as good approximate solution. However, te relative error as a sarp increase at te rigt boundary as sown in figure. We observe tat te error associated wit te numerical sceme diminises as te spatial step size, approaces zero. 83

11 5. Conclusion An implicit Finite Difference approac for solving te parabolic partial differential equation in one dimension as been discussed. We considered te Cran Nicolson sceme due its attractive properties suc as te second order accuracy in bot time and spatial dimensions and stability wic is unconditional. Tese mae it preferable to te explicit metod wic as first order accurate in time dimension and as a restricted stability. We note tat, toug implicit metod are computationally costly due to tat fact tat at eac time step, a wole system of equations is solved, tis is compensated by te accuracy and te unrestricted stability. Te Test case as sown tat te Cran Nicolson metod is a good alternative to solving parabolic DEs, were analytical solution is difficult to obtain or even impracticable to use. To furter improve on te accuracy of te numerical solution to te parabolic DE, we sall explore te avenues of wavelet approaces. References. Arivis, G., Maridais, C., & Nocetto, R. (6). A posteriori error estimates for te Cran Nicolson metod for parabolic equations. Matematics of computation, 5(54), Alan, J. () Advanced Engineering Matematics, st Edition. Harcourt Academic ress, Boston 3. Burden. L. R., Faires, J. D. and Reynolds. C. A. (98). Numerical Analysis. rindle, Weber and Scmidt, page Bruaset, M. A. and Tveito, A. (6). Numerical solution of partial differential equation on parallel computers, Springer-Verlog Berlin Heidelberg. 5. Claes, J. (9). Numerical Solution of artial Differential Equations by te Finite Element Metod. Cambridge University ress, New Yor. 6. Douglas Jr, J., & Kim, S. (). Improved accuracy for locally one-dimensional metods for parabolic equations. Matematical Models and Metods in Applied Sciences, (9), Lapidus, L. and inder, F. G. (999). Numerical solution of artial Differential Equation in Science and Engineering, University of Vermont. 8. LeVeque, J. R. (). Finite Difference Metods for Ordinary and artial Differential Equations: Steady- State and Time-Dependent roblems. SIAM, iladelpia. 9. Kersale E. (3). Analytic Solutions of artial Differential Equations. Scool of Matematics, University of Leeds. ersale.. Kincaid, D. and Ceney, W. (). Numerical Analysis: Matematics of Scientific Computing, 3rd Edition. Broos Cole. Morton, K. W. and Mayer, D. (5). Numerical Solution of artial Differential Equation Second Edition, Cambridge University ress.. Strauss, A. W. (8). artial Differential Equations: An Introduction. Brown University, Jon Wiley & Sons, Florida. 3. Süli, E. (). Finite Element Metods for artial Differential Equations. Lecture Notes. Matematical Institute University of Oxford. 84