Finite volume method for boundary value problem Discontinuous coefficients and high Peclet number

Praveen. C praveen@math.tifrbng.res.in

Tata Institute of Fundamental Research Center for Applicable Mathematics

Bangalore 560065 http://math.tifrbng.res.in/~praveen

January 21, 2013

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http://math.tifrbng.res.in/~praveen

Discontinuous coefficients

− d dx a(x)

du

dx = q, x ∈ Ω = (0, 1)

Finite difference scheme

−1 2

[D+x (aD − x ) +D

− x (aD

+ x )]uj = qj , j = 1, 2, . . . , n− 1

or, for j = 1, 2, . . . , n− 1 1

2h2 [−(aj−1 + aj)uj−1 + (aj−1 + 2aj + aj+1)uj − (aj + aj+1)uj+1] = qj

Boundary condition u(0) = 0, u(1) = 1

Discontinuous coefficient

a(x) =

{ � 0 ≤ x ≤ x∗ 1 x∗ < x ≤ 1

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Discontinuous coefficients

Jump condition: au′ is continuous across the discontinuity

� lim x↑x∗

u′(x) = lim x↓x∗

u′(x)

Exact solution (q = 0)

u(x) =

{ αx 0 ≤ x < x∗ �αx+ 1− �α x∗ ≤ x ≤ 1

where

α = 1

(1− �)x∗ + �

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Discontinuous coefficients

Finite difference solution: Consider the uniform grid xj = jh, j = 0, 1, . . . , n with h = 1n . Assume xk < x

∗ ≤ xk+1. Postulate FD solution as (u0 = 0 and un = 1 is satisfied)

uj =

{ αhj 0 ≤ j ≤ k βhj − βhn+ 1 k + 1 ≤ j ≤ n

where (apply FD equation at j = k and j = k + 1)

αh =

[ � 1− � 1 + �

+ �(n− k) + k ]−1

, βh = �αh

At x = xk = hk, FD solution is

uk = xk

�h1−�1+� + (1− �)xk + �

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Discontinuous coefficients Case x∗ = xk+1: Exact solution is

u(xk) = xk

(1− �)xk+1 + � and error in FD solution

uk − u(xk) = O ( � 1− � 1 + �

h

)

Case x∗ = xk + h/2: Exact solution is

u(xk) = xk

(1− �)xk + �+ 12h(1− �)

and error in FD solution

uk − u(xk) = O (

(1− �)2 �(1 + �)

h

) 5 / 38

Discontinuous coefficients

a(x) error

constant 0 continuous O

( h2 )

discontinuous O (h)

FD scheme loses its optimal accuracy when coefficients are discontinuous.

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Finite volume scheme

Non-overlapping finite volumes (vertex-based or vertex-centered)

Ωj = (xj − h

2 , xj +

h

2 ), j = 1, 2, . . . , n− 1

0 1 n− 1 njj − 1 j + 1

j − 1/2 j + 1/2

Ωj

x = 0 x = 1

Integrate over each cell Ωj

− ∫

Ωj

(au′)′dx =

∫ Ωj

qdx

−[(au′)j+ 1 2 − (au′)j− 1

2 ] ≈ hqj

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Finite volume scheme

uj+1 − uj = ∫ xj+1 xj

u′dx

=

∫ xj+1 xj

1

a (au′)dx

≈ (au′)j+ 1 2

∫ xj+1 xj

1

a dx

≈ (au′)j+ 1 2

h

aj+ 1 2

where 1

aj+ 1 2

= 1

2

( 1

aj +

1

aj+1

) ≈ 1 h

∫ xj+1 xj

1

a dx

Flux at x = xj+ 1 2

(au′)j+ 1 2

= aj+ 1 2

uj+1 − uj h

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Finite volume scheme Finite volume scheme

− [ aj+ 1

2

uj+1 − uj h

− aj− 1 2

uj − uj−1 h

] = hqj , j = 1, 2, . . . , n− 1

with u0 = 0, un = 1

Discontinuous coefficient, x∗ = xk + h/2: Discontinuity coinciding with cell face xk+ 1

2

aj+ 1 2

=

 �, 1 ≤ j < k 2�

1+� , j = k

1, k < j ≤ n− 1 Postulating FVM solution as before, we get

αh = h

(1− �)x∗ + � , βh = �αh 9 / 38

Finite volume scheme

The FVM solution is exact !!! In general, the error of FV solution is O ( h2 ) .

Discontinuous coefficient, x∗ = xk: Discontinuity inside a cell

aj+ 1 2

is again given as in previous case and the error in uk is O ( h2 )

(show this).

When dealing with problems with discontinuous coefficients and/or solutions, finite volume methods are more accurate than finite difference methods. FVM has better mathematical basis since it is based on weak formulation.

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If mesh Peclet number is large

µ > 0 and U > 0 are constants

−µu′′ + Uu′ = 0, x ∈ (0, L)

Non-dimensionalize: x→ x/L

− 1 Pe u′′ + u′ = 0, x ∈ (0, 1)

Peclet number:

Pe = UL

µ =

convection

diffusion

General solution

u(x) = A+BexPe

Boundary conditions

u(0) = a, u(1) = b

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If mesh Peclet number is large Solution

u(x) = a+ (b− a)e (x−1)Pe − e−Pe

1− e−Pe Example

a = 0, b = 1, u(x) = e(x−1)Pe − e−Pe

1− e−Pe

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 Pe=1

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 Pe=50

Boundary layer is present near x = 1 for large value of Pe Boundary layer thickness = O

( 1

Pe

) for Pe� 1

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If mesh Peclet number is large Finite volume scheme (cell-centered)

1 njj − 1 j + 1

j − 1/2 j + 1/2

Ωjx = 0 x = 1

Integrating over finite volume Ωj

− 1 Pe

[u′ j+ 1

2

− u′ j− 1

2

] + uj+ 1 2 − uj− 1

2 = 0

Fluxes: for j = 1, 2, . . . , n− 1

uj+ 1 2

= 1

2 (uj + uj+1), u

′ j+ 1

2

= uj+1 − uj

h

Boundary fluxes

u 1 2

= a, un+ 1 2

= b, u′1 2

= u1 − a h/2

, u′ n+ 1

2

= b− un h/2

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If mesh Peclet number is large

System of finite volume equations

(1 + 6

P )u1 + (1−

2

P )u2 = (2 +

4

P )a

−(1 + 2 P

)uj−1 + 4

P uj + (1−

2

P )uj+1 = 0, j = 2, 3, . . . , n− 1

−(1 + 2 P

)un−1 + (−1 + 6

P )un = (−2 +

4

P )b

with the mesh Peclet number P = hPe .

Case P = 0: No convection, only diffusion

uj = A+Bj

Determine A,B from first and last equations.

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If mesh Peclet number is large Case P = 2:

uj = a, j = 1, 2, . . . , n

Case P 6= 0, 2: Postulate solution of the form

uj = A+Bz j =⇒ z = 2 + P

2− P Determine A,B from first and last equations.

uj = a+ b− a zn − 1

( −1 + 2− P

2 zj )

P z Solution

0 < P < 2 z > 0 monotonic P > 2 z < 0 oscillatory

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If mesh Peclet number is large

Non-oscillatory solution is guaranteed provided h < 2Pe . If Pe� 1, then we need a very fine mesh, h� 1.

æ æ æ æ æ æ æ

æ

æ

æ

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0 Pe=10, n=10, P=1

æ æ æ æ æ æ æ

æ

æ

æ

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.5

1.0 Pe=40, n=10, P=4

Solution is oscillatory at higher Pe numbers; does not satisfy maximum principle

Matlab code: convdiff central ccfvm.m

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If mesh Peclet number is large

Adaptive grid: Boundary layer thickness

∆ = 4

Pe

Put 1/3 of total cells in the boundary layer.

Grid refined in the boundary layer

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If mesh Peclet number is large

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Oscillation 6= instability • Stability refers to influence of perturbations in data (initial and

boundary conditions) on the solution

• Consider a perturbation ∆a of the boundary value a. Then

∆uj = zn − zj + zj P2

zn − 1 ∆a

• Solution is stable in norm ‖·‖ if there is a constant C independent of h such that

‖∆u‖ ≤ C|∆a|

• Case z < −1:

‖∆u‖∞ ≤ C|∆a|, C = 2 (

2 + |P | 2

) 19 / 38

Oscillation 6= instability

• Case −1 ≤ z < 0 and perturbations in b are left as an exercise. • As h→ 0, the condition h < 2/Pe will be satisfied; the numerical

solution converges to the exact solution.

• In practical computations, we use a finite grid h. It is desirable to have a scheme free of oscillations for all values of h.

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Upwind scheme

d

dx (Uu)− d

dx D

du

dx = q

Finite volume scheme

(Uu)j+ 1 2 − (Uu)j− 1

2 − [(Du′)j+ 1

2 − (Du′)j− 1

2 ] = qjh

Central scheme for convection term

(Uu)j+ 1 2

= 1

2 Uj+ 1

2 (uj + uj+1)

Upwind scheme for convection term

(Uu)j+ 1 2

=

{ Uj+ 1

2 uj , Uj+ 1

2 ≥ 0

Uj+ 1 2 uj+1, Uj+ 1

2 < 0