Progress through quality Education Finite Difference Method

S. NADARAJA PILLAI

  

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Finite Difference Method

  School of Mechanical Engineering

  

SASTRA University

  Thanjavur – 613401 Email: nadarajapillai@mech.sastra.edu

  

@

Faculty Development Program on Computational Fluid Dynamics

June 2016

13 June 2016

  Outline Introduction Finite Difference Method Discretization Methods Forward Backward Central Difference Schemes Errors Examples

  

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  Finite Difference Method (FDM)  Historically, the oldest of the three  Techniques published as early as 1910 by L. F. Richardson  Seminal paper by Courant, Fredrichson and Lewy (1928) derived stability criteria for explicit time stepping  First ever numerical solution: flow over a circular cylinder by Thom (1933)  Scientific American article by Harlow and Fromm (1965) clearly and publicly the idea of “computer experiments” for the first time — CFD is born!!

  

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  Discretization methods

• First step in obtaining a numerical solution is to discretize the geometric domain to define a numerical grid
• Each node has one unknown and need one algebraic equation, which is a relation between the variable value at that node and those at some of the neighboring nodes.
• The approach is to replace each term of the PDE at the particular node by a finite-difference approximation.
• Numbers of equations and unknowns must be equal

  

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  Finite Difference Techniques

• Taylor Series Expansion : Any continuous differentiable function, in

  the vicinity of x , can be expressed as a Taylor series:

  i _{2 2 3 n 3 n}       ∂ Φ x − x ∂ Φ x − x ∂ Φ x − x ∂ Φ

    ( ) ( ) ( ) _{i i i} Φ ( ) x = Φ ( ) ( x x −

  ... H + + + x ) + _{i i}   _{2 3 n}       ∂ x 2 ! ∂ x 3 ! ∂ x n ! ∂ x   _i       _{i i i 2 2 3}    

  ∂ Φ Φ − Φ x − x ∂ Φ x − x ∂ Φ   ( ) _{i 1 i i} + +

  − H   _{2 3}     ∂ x x − x 2 ∂ x 6 ∂ x   _{i i 1 i}

• _{1 i i 1 i} = −
• _{i i}

     

• Higher order derivatives are unknown and can be dropped when the distance between grid points is small.
• By writing Taylor series at different nodes, x , x , or both x and

  i-1 i+1 i-1

  x , we can have:

  i+1 ∂ Φ Φ − Φ   _{i i} − ₁

  ∂ Φ Φ − Φ   _{i 1 i} ≈  

  ≈

•  

  Forward-FDS Backward-FDS ∂ x x − x   _{i i i − 1} ∂ x x − x

   

• i i ^{1 i}

  st ∂ Φ Φ − Φ   _{i 1 i} − + ₁

  1 order, order of accuracy P =1 kest

  ≈   ∂ x x − x Central-FDS  

• i i ^{1 i −}
¹ nd

  2 order, order of accuracy P =1 kest

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  Finite Difference Techniques

• • Numerical solutions can give answers at only discrete points in

  the domain, called grid points.

• • If the PDEs are totally replaced by a system of algebraic

  equations which can be solved for the values of the flow-field variables at the discrete points only, in this sense, the original PDEs have been discretized. Moreover, this method of .

  discretization is called the method of finite differences

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  Finite Difference Techniques

  A partial derivative replaced with a suitable algebraic difference quotient is • called finite difference. Most finite-difference representations of derivatives are based on Taylor’s series expansion. Taylor’s series expansion: •

  Consider a continuous function of x, namely, f(x), with all derivatives

  x ∆ x

• defined at x. Then, the value of f at a location can be estimated

  from a Taylor series expanded about point x, that is,

  n

  2

  3 f 1 f 1 f 1 f

  ∂ ∂ ∂ ∂

  2

  3 n f x x f x x x x x ( ∆ ) = ( ) ∆ ∆ ∆ ... ( ∆ ) ... + + + + + + +

  ( ) ( ) n

  2

  3 x x x n x

  ∂ 2 ! ∂ 3 ! ∂ ! ∂

• In general, to obtain more accuracy, additional higher-order terms must be included.

  

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  Finite Difference Techniques

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  Forward, Backward and Central Difference Scheme

  (1) Forward difference: Neglecting higher-order terms, we can get _{2 2 3 3 n n}

  ∂ f ( x − x ) ∂ f x − x ∂ f ( x − x ) ∂ f ( ) ⁱ

• ^{1 i i}
• ^{1 i i 1 i}
• i ^{1 i i i i}
_{2 3 n} (a)

  f ( x ) = + + _{1 i i i} f ( x ) ( ) ( x − x ) ( ) ( ) ... ( ) ... + + + +

  ∂ x 2 ! ∂ x 3 ! ∂ x n ! ∂ x ( ) ( ) ( ) ( ) ∂ f f x − f x f x − f x _i + + _{1 i i 1 i}

  ( ) ; x x x = =

  ∆ = − _i

• i
^{1 i 1 i} ∂ x x − x ∆ x _{1 i i i} + + ¹ Progress through quality Education

  13 June 2016

• − ∂ ∂ − =
• ∂ ∂ − − + + ∂ ∂ −

  − − − = ∆

∆

  Difference Scheme

  …(c) Forward, Backward and Central

  = ∂ ∂ _{i i i i i} x x x f x f x f

  − + − + − −

  1 ^{1 1 1} ) ( ) ( ) (

  (b)

  = ∂ ∂ _{i i i i i i i i i i i} x x x

x

x f x f x x x f x f x f

  

−

= − −

  ) ( − −

  

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  1 ¹ ₁ ¹ ; ) ( ) ( ) ( ) (

  − − − − − ⁱ _n ^{n n i i n i i i}

ⁱ

^{i i i i i i i} x f n x x x f x x x f x x x x x f x f x f

  − ∂ ∂ −

  2 ) ( ) ( ) ( ) ( ) ( ¹ ₃ ^{3 3 1} ₂ ^{2 2 1} _{1 1}

  3 ) ( !

  ) ( ) ... 1 ( ) ( !

  ( ) ... ) ( !

  (2) Backward difference: Neglecting higher-order terms, we can get (3) Central difference: (a)-(b) and neglecting higher-order terms, we can get

13 June 2016

  

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  Forward, Backward and Central Difference Scheme

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  Truncation Error

  Truncation error: • The higher-order term neglecting in Eqs. (a), (b), (c) constitute the truncation error. The general form of Eqs. (d), (e), (f) plus truncated terms can be written as

  f f f

  ∂ −

  1 i

o x

  ( ) = ( ∆ ) Forward:

• i

  i x x

  ∂ ∆

  f f f

  ∂ − Backward:

  i i −

  1

o x

  ( ) = ( ∆ + )

  i x x

  ∂ ∆

  f f f

  ∂ −

  2 i 1 i −

1 Central:

  o x

  ( ) = ( ∆ )

  i x x

  ∂ 2 ∆

  

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• Second derivatives:
• Central difference:
• 1
• − =

• Forward difference:
• Backward difference:
• − =
• − =

  1

  2

  2

  2 x o x f f f x f i i i i

  ∆ + ∆

  ∂ ∂

  ) ( ) (

  2

  2 ) (

  2

  1

  2

  2 x o x f f f x f i i i i

  ∆ + ∆

  ∂ ∂

  − − Second Order Derivatives

  2

  2 ) (

  ) ( ) (

  1

  

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  If , then (a)+(b) becomes

  x x x _{i i}

  ∆ = ∆ = ∆

  2

  2

  1

  − +

  2

  2

  ) ( ) (

  2 ) (

  x o x f f f x f i i i i

  ∆ + ∆

  ∂ ∂

13 June 2016

  Mixed Derivatives

  Mixed derivatives: •

• Taylor series expansion:
_{2 2 2 2 2}

  ∂ f ∂ f ( ∆ x ) ∂ f ( ∆ y ) ∂ f ( ∆ x )( ∆ y ) ∂ f ³

³

  2 [( ∆ ) , ( ∆ ) ] f x x y y f x y x y _{2 2} o x y

• ( ∆ , ∆ ) = ( , ) ∆ ∆

  ∂ x ∆ y

2 ! ∂ x

2 ! ∂ y 2 ! ∂ x ∂ y
• Central difference:
₂

  − + + − − − + + ^{1 , 1 1 , 1 1 , 1} _{2 2} =

  f f − f f − +   ∂ f ^{i j i j i j i j} _{1 , 1}

• o ∆ x ∆ y

  [( ) , ( ) ]   x y x y

  ∂ ∂ 4 ( ∆ )( ∆ )   i j _,

• Forward difference:
₂

  f − f − + f f   ∂ f ^{i j i j i j i j} _{1 ,}

• ^{1 ,}
^{1 1 , ,} = o ∆ x ∆ + y [( ) , ( ) ]

    x y x y

  ∂ ∂ ( ∆ )( ∆ )   i j _,

• Backward difference:

  f − f − f f   ∂ f

• ²
• = o ∆ x ∆ y ^{i j i j i j i j , − 1 , , − 1 − 1 , −}

¹

[( ) , ( ) ]   x y x y

  ∂ ∂ ( ∆ )( ∆ )   i j _,

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  Errors Involved

  In the solution of differential equations with finite differences, a variety of • schemes are available for the discretization of derivatives and the solution of the resulting system of algebraic equations. In many situations, questions arise regarding the •

  round-off and truncation

errors involved in the numerical computations, as well as the consistency,

  stability and the convergence of the finite difference scheme. Round-off errors:computations are rarely made in exact arithmetic. This • means that real numbers are represented in “floating point” form and as a result, errors are caused due to the rounding-off of the real numbers. In extreme cases such errors, called “round-off” errors, can accumulate and become a main source of error.

  

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  Errors Involved

  Truncation error: In finite difference representation of derivative with • Taylor’s series expansion, the higher order terms are neglected by truncating the series and the error caused as a result of such truncation is called the “truncation error”.

  The truncation error identifies the difference between the exact solution of • a differential equation and its finite difference solution without round-off error.

  

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THANK YOU

  

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