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Multi-domain Bivariate Spectral Local Linearisation method for solving non-similar boundary layer partial differential equations

MAGAGULA VUSI MPENDULO Supervisors: Prof. S.S. Motsa & Prof. P. Sibanda The 40th South African Symposium of Numerical and Applied Mathematics

22 - 24 March 2016

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Overview

1

AIM

2

SOLUTION PROCEDURE

3

RESULTS AND DISCUSSION

4

CONCLUSION & FUTURE RESEARCH DIRECTION

5

REFERENCES

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AIM

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.

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SOLUTION PROCEDURE

Solution Method

The solution approach involves Domain decomposition linearisation and decoupling bivariate interpolation. pseudo-spectral approximation

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SOLUTION PROCEDURE

Numerical Experiment

∂3f ∂η3 + 1 4(n + 3)f ∂2f ∂η2 − 1 2(n + 1) ∂f ∂η 2 + ζ ∂2f ∂η2 + (1 − w)g + wh = 1 4(1 − n)ζ ∂f ∂η ∂2f ∂ζ∂η − ∂2f ∂η2 ∂f ∂ζ

• ,

(1) 1 Pr ∂2g ∂η2 + 1 4(n + 3)f ∂g ∂η + ζ ∂g ∂η = 1 4(1 − n)ζ ∂f ∂η ∂g ∂ζ − ∂g ∂η ∂f ∂ζ

• ,

(2) 1 Sc ∂2h ∂η2 + 1 4(n + 3)f ∂h ∂η + ζ ∂h ∂η = 1 4(1 − n)ζ ∂f ∂η ∂h ∂ζ − ∂h ∂η ∂f ∂ζ

• ,

(3) subject to f(ζ, 0) = 0, ∂f ∂η (ζ, 0) = 0, g(ζ, 0) = h(ζ, 0) = 1, ∂f ∂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

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SOLUTION PROCEDURE

Domain Decomposition

ζ0 ζ1 ζ2 ζ3 ζl−1 ζl ζp−1 ζp Γ1 Γ2 Γ3 Γl Γp ζl−1 ζl ζ(l) ζ(l)

1

ζ(l)

2

ζ(l)

s−1

ζ(l)

s

Figure: Multi-domain Grid The patching condition requires that f (l)(η, ζl−1) = f (l−1)(η, ζl−1), η ∈ [a, b], (4) where f (l)(η, ζ) denotes the solution of equation (2) at each sub-interval Γl

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SOLUTION PROCEDURE

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 [ζl−1, ζl], these decoupled linear PDEs can be solved iteratively in a sequential manner until the desired solution is obtained. β0,r ∂3f (l)

r+1

∂η3 + β1,r ∂2f (l)

r+1

∂η2 + β2,r ∂f (l)

r+1

∂η + β3,rf (l)

r+1 + β4,r

∂f (l)

r+1

∂ξ + β5,r ∂f ′(l)

r+1

∂ξ = R(l)

f,r,

(5) σ1,r ∂2g(l)

r+1

∂η2 + σ2,r ∂g(l)

r+1

∂η + σ3,rg(l)

r+1 + σ4,r

∂g(l)

r+1

∂ξ = R(l)

g,r,

(6) ω1,r ∂2h(l)

r+1

∂η2 + ω2,r ∂h(l)

r+1

∂η + ω3,rh(l)

r+1 + ω4,r

∂h(l)

r+1

∂ξ = R(l)

h,r,

(7)

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SOLUTION PROCEDURE

Pseudo-spectral Approximation

Assume that the solution at each sub-interval Γl, denoted by f (l)(η, ζ), can be approximated by a bivariate Lagrange interpolation polynomial of the form f (l)(η, ζ) ≈

Nx

• i=0

Nt

• j=0

f (l)(ηi, ζj)Li(η)Lj(ζ). (8) Transpiration parameter derivative values are computed at the grid points (ηi, ζj): ∂f (l) ∂ζ

• (ηi,ζj)

=

• 2

ζl − ζl−1 Nt

• v=0

djvf (l)(ηj, ζv) =

• 2

ζl − ζl−1 Nt

• v=0

djvF(l)

v

(9) The nth order space derivative is defined as ∂nf (l) ∂ηn

• (ηi,ζj)

= 2 η∞ n

Nx

• ρ=0

Dn

iρf (l)(ηρ, ζj) =

2 η∞ n DnF(l)

j ,

j = 0, 1, 2, . . . , Nt, (10) where the vector F(l)

j

is defined as F(l)

j

= [f (l)(η0, ζj), f (l)(η1, ζj), . . . , f (l)(ηNx, ζj)]T. (11)

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SOLUTION PROCEDURE

Pseudo-spectral Approximation

Substituting equations (9) and (10) into equation (5), we get

• β

β β0,rD3 + β β β1,rD2 + β β β2,rD + β β β3,r

• F(l)

r+1,j + β

β β4,r

Nt

• v=0

djvF(l)

r+1,v + β

β β5,r

Nt

• v=0

djvDF(l)

r+1,v = R(l) f,r,

(12) for j = 0, 1, 2, . . . , Nt. The patching condition requires that f (l)

r+1(ηi, ζ(l−1,j)) = f (l−1) r+1 (ηi, ζ(l−1,j)),

η ∈ [a, b], (13) The initial unsteady solution of equation (5) when ζ = 0 corresponds to t = tNt = −1. Equation (12) is evaluated for j = 0, 1, · · · , Nt − 1

• β

β β0,rD3 + β β β1,rD2 + β β β2,rD + β β β3,r

• F(l)

r+1,j + β

β β4,r

Nt−1

• v=0

djvF(l)

r+1,v + β

β β5,r

Nt−1

• v=0

djvDF(l)

r+1,v = R

R R(l)

1,j, (14)

and R R R(l)

1,j = R(l) f,r − β

β β4,rdjNtF(l)

Nt − β

β β5,rdjNtDF(l)

Nt .

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SOLUTION PROCEDURE

Imposing boundary conditions for j = 0, 1, · · · , Nt − 1, equation (14) can be expressed as the following Nt(Nx + 1) × Nt(Nx + 1) matrix system      A0,0 A0,1 · · · A0,Nt−1 A1,0 A1,1 · · · A1,Nt−1 . . . . . . ... . . . ANt−1,0 ANt−1,1 · · · ANt−1,Nt−1            F(l) F(l)

1

. . . F(l)

Nt−1

      =       R R R(l)

1,0

R R R(l)

1,1

. . . R R R(l)

1,Nt−1

      , (15) where Ai,i = β β β0,rD3 + β β β1,rD2 + β β β2,rD + β β β3,rI + β β β4,rdiiI + β β β5,rdiiD (16) Ai,j = β β β4,rdijI + β β β5,rdijD, when i = j, (17)

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RESULTS AND DISCUSSION

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 ζ f ′′(0, ζ) f ′′(0, 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: Nx = 60, Nt = 5, p = 20

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RESULTS AND DISCUSSION

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 ζ −h′(0, ζ) −h′(0, 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: Nx = 60, Nt = 5, p = 20

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RESULTS AND DISCUSSION

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: Nx = 60, Nt = 5, P = 20

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CONCLUSION & FUTURE RESEARCH DIRECTION

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.

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REFERENCES

References

Motsa, S.S. and Animasaun, I.L., 2015. A new numerical investigation of some thermo-physical properties on unsteady MHD non-Darcian flow past an impulsively started vertical surface. Thermal Science, 19(suppl. 1), 249-258. Hussain S., Hossain M.A., Natural convection flow from a vertical permeable flat plate with variable surface temperature and species concentration, Engineering Computations,Vol. 17 No. 7,2000, 789-812 Trefethen, Lloyd N. Spectral methods in MATLAB. Vol. 10. Siam, 2000. Canuto C., Hussaini M. Y., Quarteroni A., and T. A. Zang, Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin, 2007.

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REFERENCES Vusi Magagula Multi-domain BSLLM 22 - 24 March 2016 16 / 16