Multi-domain Bivariate Spectral Local Linearisation method for solving non-similar boundary layer partial differential equations M AGAGULA V USI M PENDULO Supervisors: Prof. S.S. Motsa & Prof. P. Sibanda The 40 th South African Symposium of Numerical and Applied Mathematics 22 - 24 March 2016 Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 1 / 16
Overview A IM 1 S OLUTION P ROCEDURE 2 R ESULTS AND D ISCUSSION 3 C ONCLUSION & F UTURE R ESEARCH D IRECTION 4 R EFERENCES 5 Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 2 / 16
A IM Aim & Objectives Aim To present an extension of the Bivariate Spectral Local Linearisation Method (BSLLM)[1] for solving non-similar nonlinear PDEs over large time intervals . BSLLM uses Chebyshev-Gauss-Lobbatto points (see [3, 4]) Drawback of BSLLM - accuracy deteriorates over large time intervals . New approach termed - Multi-domain Bivariate Spectral Local Linearisation Method (MD-BSLLM) Objectives Solve non-linear non-similar boundary layer equations over a large time domain using the MD-BSLLM. Validate the results using a series solution approach. Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 3 / 16
S OLUTION P ROCEDURE Solution Method The solution approach involves Domain decomposition linearisation and decoupling bivariate interpolation. pseudo-spectral approximation Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 4 / 16
S OLUTION P ROCEDURE Numerical Experiment � ∂ f � 2 ∂ 3 f 4 ( n + 3 ) f ∂ 2 f + ζ ∂ 2 f ∂η 3 + 1 ∂η 2 − 1 2 ( n + 1 ) ∂η 2 + ( 1 − w ) g + wh ∂η � ∂ f ∂ζ∂η − ∂ 2 f ∂ 2 f � = 1 ∂ f 4 ( 1 − n ) ζ , (1) ∂η ∂η 2 ∂ζ � ∂ f ∂ 2 g � 1 ∂η 2 + 1 4 ( n + 3 ) f ∂ g ∂η + ζ ∂ g ∂η = 1 ∂ g ∂ζ − ∂ g ∂ f 4 ( 1 − n ) ζ , (2) Pr ∂η ∂η ∂ζ � ∂ f ∂ 2 h 4 ( n + 3 ) f ∂ h ∂η + ζ ∂ h ∂ h ∂ζ − ∂ h ∂ f � 1 ∂η 2 + 1 ∂η = 1 4 ( 1 − n ) ζ , (3) Sc ∂η ∂η ∂ζ subject to ∂ f ∂ f f ( ζ, 0 ) = 0 , ∂η ( ζ, 0 ) = 0 , g ( ζ, 0 ) = h ( ζ, 0 ) = 1 , ∂η ( ζ, ∞ ) = g ( ζ, ∞ ) = h ( ζ, ∞ ) = 0 . Formulated and Solved by Hussain et.al. [2] using finite-difference based Keller-box technique Results were validated using Series Solutions , for small and large ζ Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 5 / 16
S OLUTION P ROCEDURE Domain Decomposition Γ p Γ 1 Γ 2 Γ 3 Γ l ζ 0 ζ 1 ζ 2 ζ 3 ζ l − 1 ζ l ζ p − 1 ζ p ζ l − 1 ζ l ζ ( l ) ζ ( l ) ζ ( l ) ζ ( l ) ζ ( l ) s 0 1 2 s − 1 Figure: Multi-domain Grid The patching condition requires that f ( l ) ( η, ζ l − 1 ) = f ( l − 1 ) ( η, ζ l − 1 ) , η ∈ [ a , b ] , (4) where f ( l ) ( η, ζ ) denotes the solution of equation (2) at each sub-interval Γ l Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 6 / 16
S OLUTION P ROCEDURE Linearisation and Decoupling The PDEs are linearised using the quasi-linearisation method . an iterative procedure based on the Taylor-series expansion about the previous estimates of the solution. Applying the quasi-linearisation method on one function at a time, results in three decoupled linear PDEs . In each sub-interval [ ζ l − 1 , ζ l ] , these decoupled linear PDEs can be solved iteratively in a sequential manner until the desired solution is obtained. ∂ 3 f ( l ) ∂ 2 f ( l ) ∂ f ( l ) ∂ f ( l ) ∂ f ′ ( l ) r + 1 r + 1 r + 1 + β 3 , r f ( l ) r + 1 r + 1 = R ( l ) β 0 , r + β 1 , r + β 2 , r r + 1 + β 4 , r + β 5 , r f , r , (5) ∂η 3 ∂η 2 ∂η ∂ξ ∂ξ ∂ 2 g ( l ) ∂ g ( l ) ∂ g ( l ) + σ 3 , r g ( l ) r + 1 r + 1 r + 1 = R ( l ) + σ 2 , r r + 1 + σ 4 , r σ 1 , r g , r , (6) ∂η 2 ∂η ∂ξ ∂ 2 h ( l ) ∂ h ( l ) ∂ h ( l ) r + 1 r + 1 + ω 3 , r h ( l ) r + 1 = R ( l ) ω 1 , r + ω 2 , r r + 1 + ω 4 , r h , r , (7) ∂η 2 ∂η ∂ξ Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 7 / 16
S OLUTION P ROCEDURE Pseudo-spectral Approximation Assume that the solution at each sub-interval Γ l , denoted by f ( l ) ( η, ζ ) , can be approximated by a bivariate Lagrange interpolation polynomial of the form N x N t � � f ( l ) ( η, ζ ) ≈ f ( l ) ( η i , ζ j ) L i ( η ) L j ( ζ ) . (8) i = 0 j = 0 Transpiration parameter derivative values are computed at the grid points ( η i , ζ j ) : � N t � N t ∂ f ( l ) � � � 2 2 � d jv f ( l ) ( η j , ζ v ) = � d jv F ( l ) � = (9) � v ∂ζ ζ l − ζ l − 1 ζ l − ζ l − 1 � ( η i ,ζ j ) v = 0 v = 0 The n th order space derivative is defined as � 2 � 2 N x ∂ n f ( l ) � n � n � � D n F ( l ) � D n i ρ f ( l ) ( η ρ , ζ j ) = = j , j = 0 , 1 , 2 , . . . , N t , (10) � ∂η n η ∞ η ∞ � ( η i ,ζ j ) ρ = 0 where the vector F ( l ) is defined as j F ( l ) = [ f ( l ) ( η 0 , ζ j ) , f ( l ) ( η 1 , ζ j ) , . . . , f ( l ) ( η N x , ζ j )] T . (11) j Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 8 / 16
S OLUTION P ROCEDURE Pseudo-spectral Approximation Substituting equations (9) and (10) into equation (5), we get N t N t � β 0 , r D 3 + β β 1 , r D 2 + β � F ( l ) � d jv F ( l ) � d jv DF ( l ) r + 1 , v = R ( l ) β 2 , r D + β r + 1 , j + β r + 1 , v + β β β β β β 3 , r β β β 4 , r β β 5 , r f , r , (12) v = 0 v = 0 for j = 0 , 1 , 2 , . . . , N t . The patching condition requires that f ( l ) r + 1 ( η i , ζ ( l − 1 , j ) ) = f ( l − 1 ) r + 1 ( η i , ζ ( l − 1 , j ) ) , η ∈ [ a , b ] , (13) The initial unsteady solution of equation (5) when ζ = 0 corresponds to t = t N t = − 1. Equation (12) is evaluated for j = 0 , 1 , · · · , N t − 1 N t − 1 N t − 1 � β 0 , r D 3 + β β 1 , r D 2 + β � F ( l ) d jv F ( l ) d jv DF ( l ) R ( l ) � � β β β β β 2 , r D + β β 3 , r β r + 1 , j + β β β 4 , r r + 1 , v + β β β 5 , r r + 1 , v = R R 1 , j , (14) v = 0 v = 0 and R ( l ) 1 , j = R ( l ) β 4 , r d jN t F ( l ) β 5 , r d jN t DF ( l ) R R f , r − β β N t − β β N t . Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 9 / 16
S OLUTION P ROCEDURE Imposing boundary conditions for j = 0 , 1 , · · · , N t − 1, equation (14) can be expressed as the following N t ( N x + 1 ) × N t ( N x + 1 ) matrix system R ( l ) F ( l ) R R · · · A 0 , 0 A 0 , 1 A 0 , N t − 1 0 1 , 0 F ( l ) R ( l ) · · · A 1 , 0 A 1 , 1 A 1 , N t − 1 R R 1 1 , 1 = , (15) . . . ... . . . . . . . . . . . . A N t − 1 , 0 A N t − 1 , 1 · · · A N t − 1 , N t − 1 F ( l ) R ( l ) R R N t − 1 1 , N t − 1 where β 0 , r D 3 + β β 1 , r D 2 + β A i , i = β β β β 2 , r D + β β β β 3 , r I + β β β 4 , r d ii I + β β β 5 , r d ii D (16) A i , j = β β β 4 , r d ij I + β β β 5 , r d ij D , when i � = j , (17) Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 10 / 16
R ESULTS AND D ISCUSSION Results Comparison of Multi-domain bivariate spectral local linearisation solution for the skin friction against the series solution for large ζ . MD-BSLLM Series Solution for large ζ f ′′ ( 0 , ζ ) f ′′ ( 0 , ζ ) ζ 5 0.3088214 0.3088214 10 0.1547399 0.1547399 15 0.1031717 0.1031717 20 0.0773803 0.0773803 25 0.0619045 0.0619045 30 0.0515872 0.0515872 35 0.0442176 0.0442176 40 0.0386905 0.0386905 Table: N x = 60, N t = 5, p = 20 Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 11 / 16
R ESULTS AND D ISCUSSION Results Comparison of Multi-domain bivariate spectral local linearisation solution for the Sherwood number against the series solution for large ζ MD-BSLLM Series Solution for large ζ − h ′ ( 0 , ζ ) − h ′ ( 0 , ζ ) ζ 5 3.0018658 3.0018658 10 6.0002332 6.0002332 15 9.0000691 9.0000691 20 12.0000292 12.0000292 25 15.0000149 15.0000149 30 18.0000086 18.0000086 35 21.0000054 21.0000054 40 24.0000036 24.0000036 Table: N x = 60, N t = 5, p = 20 Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 12 / 16
R ESULTS AND D ISCUSSION Results Comparison of Multi-domain bivariate spectral local linearisation solution for the Nusselt number against the series solution for large ζ MD-BSLLM Series Solution for large ζ ζ − g ′ ( 0 , ζ ) − g ′ ( 0 , ζ ) 5 3.5018961 3.5018961 10 7.0002370 7.0002370 15 10.5000702 10.5000702 20 14.0000296 14.0000296 25 17.5000152 17.5000152 30 21.0000088 21.0000088 35 24.5000055 24.5000055 40 28.0000037 28.0000037 Table: N x = 60, N t = 5, P = 20 Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 13 / 16
C ONCLUSION & F UTURE R ESEARCH D IRECTION Conclusion The MD-BSLLM method can be used to solve non-linear non-similar boundary layer equations over a large time domain. We were able to validate the results using a series solution approach. Future Research Direction Solve different types of NPDEs arising from Physics, Mathematical Biology, etc. Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 14 / 16
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