BASICS OF NUMERICS OF CFD: DISCRETIZATION METHODS

1
BASICS OF NUMERICS OF CFD DISCRETIZATION
METHODS

• Patrick F. Mensah Sr. PhD

2
Outline of Lecture

• Introduction
• Introductions to Finite Differences
• Difference Equations
• Explicit and Implicit Approaches
• Introductions to Finite Control Volume Approach
• Error and an Analysis of Stability
• Summary

3
Introduction

• Replacing of PDE or integrals in the governing
  equations with discrete numbers
• Discretization of PDE-finite differences
• Discretization of integral form-finite volumes
• Transformation of coordinate systems and grid
  system

4
Computational Domain-Rectangular Coordinates
5
Introduction

• Road map for descritization of PDEs

6
Review of Taylor Series Approximation

• Taylor series expansion of a function f(x)

First guess (not very good)
Add to capture slope
Add to account for curvature
7
Introduction to Finite Differences

• Interest is to replace PDE with suitable
  algebraic difference quotient
• Based on Taylors series expansions
• Example if ui,j denotes the x component of
  velocity at a point (i,j)

8
Introduction to Finite Differences

• Solving eqn (1) for to obtain first order
  accurate difference equation

Finite- Difference representation
Truncation error
9
Introduction to Finite Differences

• Subtracting eqn (2) from (1) for to obtain
  second order accurate difference equation for

10
Finite Difference Equations

• Summary of First Derivatives
• Two-Point Formulas-y derivative

11
Finite Difference EquationsSecond Derivatives

• Summing of Taylor series expansions (1) and (2)
  we have
• Solving for and extending to

12
Finite Difference EquationsMixed derivatives

• Taking derivative of eqns 2 and 3 with respect to
  y and subtracting them

13
Finite Difference EquationsMixed derivatives

• Solving for the mixed derivative term

14
Finite Difference Expressions with their
Finite-difference Modules
15
Finite Difference Expressions with their
Finite-difference Modules
16
Finite Difference Expressions with their
Finite-difference Modules
17
Finite difference quotient at boundaries

• For one direction polynomial approach

18
Example 4.2 Text

• See text

19
Example

• FDM for simple 1-D equation

20
Explicit and Implicit Approaches Definitions and
Contrast

• Time marching solution of parabolic equation

21
Explicit and Implicit Approaches Definitions and
Contrast

• Explicit Scheme

22
Finite Difference Representations of Parabolic
Equations and their Solutions

• Consider the basic mathematical model for a
  parabolic PDE
• the following parabolic equation

23
Finite Difference Representations of Parabolic
Equations and their Solutions

• Boundary Conditions
• Initial Conditions

24
Finite Difference Representations of Parabolic
Equations and their Solutions

• In two or three space dimensions, analogous
  equations apply

25
Computational Domain- Note for 1-D y axis
represent time and ?y?t
26
Finite Difference Representations of Parabolic
Equations and their Solutions

• Explicit Method
• Implicit Method
• A Generalization-The Theta Method
• The Crank-Nicholson Method
• Parabolic Equations in Two or Three Dimensions
• Nonlinear Parabolic Equations

27
Finite Difference Representations of Parabolic
Equations Explicit Method

• Simplest way to represent PDE as a difference
  equation
• A central/forward difference replaces the time
  derivative
• A central difference replaces the space
  derivative
• Limitation on size of time step to avoid
  instability

28
Finite Difference Representations of Parabolic
Equations Explicit Method

• xo0, x1?x,,xii?x,,xm m?x l
• to0, t1?t,,tii?t,,tnn?t T
• Using central difference approximations

29
Finite Difference Representations of Parabolic
Equations Explicit Method

• Substituting FDE into parabolic PDE and
  simplifying
• Permit calculation of dependent variable at next
  time step from known values at current and
  previous time step

30
Finite Difference Representations of Parabolic
Equations Explicit Method

• Method is unstable because of negative term on
  RHS and in general
• Numerical procedure will be stable if p, q, and
  r are positive and pqr1

31
Finite Difference Representations of Parabolic
Equations Explicit Method

• For stability a forward difference formula is
  used for the time derivative

32
Finite Difference Representations of Parabolic
Equations Explicit Method

• Stability condition

33
Grid Points of Implicit and Generalized Methods
34
Finite Difference Representations of Parabolic
Equations Implicit Method

• Finite-difference representation of Implicit
  Scheme
• At each time level algebraic equations are to be
  solve simultaneously to determine nodal values of
  the dependent variable at the next time level
• Unconditionally stable

35
Finite Difference Representations of Parabolic
Equations A Generalization-The Theta Method
36
Finite Difference Representations of Parabolic
Equations A Generalization-The Theta Method

• ? is a weighting parameter 0 ? 1
• with ?0 we get the explicit methodO?t, (?x)2
• with ?0.5 we get the Crank-Nicholson method
  O(?t)2, (?x)2
• with ?1 we get the implicit method O?t, (?x)2

37
Finite Difference Representations of Parabolic
Equations Variable weighted implicit formula
38
Finite Difference Representations of Parabolic
Equations A Generalization-The Theta Method with
BC
39
Finite Difference Representations of Parabolic
Equations A Generalization-The Theta Method
40
Finite Difference Representations of Parabolic
Equations A Generalization-The Theta Method
41
Finite Difference Representations of Parabolic
Equations The Crank-Nicholson Method

• Overcomes the limitations that the explicit
  method imposes on the size of the time steps and
  improves accuracy at the expense of having to
  solve a set of equations at each time step.
• System of equations solved is tri-diagonal for
  1-D space
• Unconditionally stable - any time step can be
  used

42
Finite Difference Representations of Parabolic
Equations The Crank-Nicholson Method

• With ?1/2

43
Finite Difference Representations of Parabolic
Equations Example

• Known initial temperature of a metal rod of
  length 1 m as follows T(x,0)x2, 0ltxlt1
• Boundary conditions at x0 and x1 are given by
  T(0,t), T(1,t)1, tgt0
• Using a20.75, ?x0.2 and ?t0.02
• Determine the temperature distribution in the rod
  for 0t1 using the explicit method

44
Finite Difference Representations of Parabolic
Equations Example

• Solution
• Governing equation
• Explicit method

45
Finite Difference Representations of Parabolic
Equations Method of Lines

• PDE is converted to a system of 1st order ODE
• The spatial derivative term is approximated with
  a finite difference formula
• The time derivative remains unchanged

46
Finite Difference Representations of Parabolic
Equations Parabolic Equations in Two or Three
Dimensions
47
Alternating-Direction Implicit Method
48
Nonlinear Parabolic Equations
49
Nonlinear Parabolic Equations

• Governing Finite Difference Equation
• Base on Alternating Direct Implicit Finite
  Difference Formulation
• Nonlinear terms (k, Cp) linearized using Taylor
  Series approximations
• Time derivative approximated using forward
  difference formulation

50
Explicit and Implicit Approaches Definitions and
Contrast

• Implicit Scheme

51
Explicit and Implicit Approaches Definitions and
Contrast

• Implicit Scheme results in a tri-diagonal
  (banded) matrix system of equations

52
DIFFERENCE EQUATIONS

• ELLIPTIC EQUATIONS

53
Finite Difference Representations of Elliptic
Equations

• Consider steady-state heat equation and its
  finite difference approximation

54
Discretization Using The Finite-Volume Method

• Use is made of the integral form of conservation
  equations and integrated over each volume

55
Assembly of Discrete System and Application of
Boundary Conditions
56
Solution of Discrete System