Analysis of Discritization Error for Finite Differences

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Analysis of Discritization Error for Finite
Differences

• (Naveed Iqbal)

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Contents

• Numerical Methods
• Computational Solution Procedures
• Discretization
• Finite Difference
• General form of PDE
• Solving Laplaces or Poissons
• Accuracy and Stability of FD Solutions
• Poisson Equation in 2D
• Error Analysis Truncation Error
• Discritization Error and Convergence

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Numerical Methods

• Objective
• Speed, Accuracy at minimum cost
• Numerical Accuracy (error analysis)
• Numerical Stability (stability analysis)
• Numerical Efficiency (minimize cost)
• Validation (model/prototype data, field data,
  analytic solution, theory, asymptotic solution)
• Reliability and Flexibility (reduce preparation
  and debugging time)
• Flow Visualization (graphics and animations)

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computational solution procedures
Governing Equations
System of Algebraic Equations
Equation (Matrix) Solver
ApproximateSolution
Discretization
Ui (x,y,z,t) p (x,y,z,t) T (x,y,z,t) or ?
(?,?,?,? )
Continuous Solutions
Finite-Difference Finite-Volume Finite-Element
Spectral Boundary Element Hybrid
Discrete Nodal Values
Tridiagonal ADI SOR Gauss-Seidel
Multigrid
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Discretization

• Spatial derivatives
• - Finite-difference Taylor-series
  expansion
• - Finite-element low-order shape function
  and
• interpolation function, continuous
  within each
• element
• - Finite-volume integral form of PDE in
  each
• control volume
• - There are also other methods, e.g.
  collocation,
• spectral method, spectral element, panel
• method, boundary element method

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Finite Difference
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Finite Difference

• Taylor series
• Truncation error
• How to reduce truncation errors?
• Reduce grid spacing, use smaller ?x x-xo
• Increase order of accuracy, use larger n

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General form of PDE

• General Equation is of form
• a ? 2? /? x2 b ? 2? /? x? y c ? 2? /? y2
  f(x,y,t)
• b2 gt 4ac Hyperbolic Equations
• b2 4ac Parabolic Equations
• b2 lt 4ac Elliptic Equations
• BoundaryEquation
  Type Curve Conditions Example
• Hyperbolic Open Cauchy Wave
  Equation
• Parabolic Open Dirichlet
  Diffusion Equation
• or
  Neumann
• Elliptic Closed Dirichlet
  Poisson Equation
• or Neumann

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Examples

• Elliptic equations
• Poisson equation,
• Parabolic equations
• Heat equations,
• Hyperbolic equations
• Conservation laws, .

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Solving Laplaces or Poissons Equation

• As always, we must convert continuous equations
  to a discrete form by setting up a mesh of points
  this is finite difference method
• h is step size in picture
• Nx grid points in x Ny in y direction
  illustrated by Nx Ny 14

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Potential in a Vacuum Filled Rectangular Box

• So imagine the worlds simplest problem
• Find the electrostatic potential inside a box
  whose sides are at a given potential
• Set up a 16 by 16 Grid on which potential defined
  and which must satisfy Laplaces Equation

? 2? /? x2 ? 2? /? y2 0
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Basic Numerical Algorithm
?Up
? middle

• Using standardcentral differencingtechniques,
  one can approximate

? ight
?Left
?Down
? 2 ? (? Left ? Right ? Up ? Down 4 ?
Middle ) / h2
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Setup for simple 16 by 16 Grid
14 by 14InternalGrid with typical local
operator characteristic of differentials
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Accuracy and Stability of FD Solutions

• The question of accuracy and stability of
  numerical methods is extremely important if our
  solution is to be reliable and useful. Accuracy
  has to do with the closeness of the approximate
  solution to exact solutions (assuming they
  exist). Stability is the requirement that the
  scheme does not increase the magnitude of the
  solution with increase in time.

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Accuracy and Stability of FD Solutions

• There are three sources of errors that are
  nearly unavoidable in numerical solution of
  physical problems
• modeling errors
• truncation (or discretization) errors, and
• round off errors.
• Each of these error types will affect accuracy
  and therefore degrade the solution.

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Accuracy and Stability of FD Solutions

• Modelling Error
• The modeling errors are due to several
  assumptions made in arriving at the mathematical
  model. For example, a nonlinear system may be
  represented by a linear PDE.
• Truncation errors
• Truncation errors arise from the fact that in
  numerical analysis, we can deal only with a
  finite number of terms from processes which are
  usually described by infinite series. For
  example, in deriving finite difference schemes,
  some higher-order terms in the Taylor series
  expansion were neglected, thereby introducing
  truncation error.

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Accuracy and Stability of FD Solutions

• Truncation errors may be reduced by using finer
  meshes, that is, by reducing the mesh size h and
  time increment ?t. Alternatively, truncation
  errors may be reduced by using a large number of
  terms in the series expansion of derivatives,
  that is, by using higher-order approximations.
  However, care must be exercised in applying
  higher order approximations. Instability may
  result if we apply a difference equation of an
  order higher than the PDE being examined.

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Accuracy and Stability of FD Solutions

• Round off errors reflect the fact that
  computations can be done only with a finite
  precision on a computer. This unavoidable source
  of errors is due to the limited size of registers
  in the arithmetic unit of the computer. Round off
  errors can be minimized by the use of
  double-precision arithmetic. The only way to
  avoid round off errors completely it to code all
  operations using integer arithmetic. This is
  hardly possible in most practical situations.

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Accuracy and Stability of FD Solutions

• Although it has been noted that reducing the mesh
  size h will increase accuracy, it is not possible
  to indefinitely reduce h. Decreasing the
  truncation error by using a finer mesh
• may result in increasing the round off error
  due to the increased number of
• arithmetic operations.

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Error as a function of the mesh size.
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Accuracy and Stability of FD Solutions
The concern about accuracy leads us to question
whether the finite difference solution can grow
unbounded, a property termed the instability of
the difference scheme. A numerical algorithm is
said to be stable if a small error at any stage
produces a smaller cumulative error. It is
unstable otherwise. To determine whether a finite
difference scheme is stable, we define an error,
?n, which occurs at time step n, assuming that
there is one independent variable.
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• We define the amplification of this error at time
  step ?n1 as

where g is known as the amplification factor. In
more complex situations, we have two or more
independent variables,
where G is the amplification matrix. For the
stability of the difference scheme, it is
required that
or
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Poisson Equation in 2D Approximation
For example
for
small
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Poisson Equation in 2D Error Analysis Truncation
Error
For
for all
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Poisson Equation in 2D Error Analysis
Stability
Error equation
If
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Poisson Equation in 2D Error Analysis
Stability
It can be shown that
Ingredients

• Positivity of the coeffiicients of
• Bound on the maximum row sum

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Discritization Error and Convergence
Job is submitted for 4 hours with some boundry
conditions
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Discritization Error and Convergence
Job is submitted for 2 hours only
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References

• Computing for Numerical Methods using Visual C
  by Shaharuddin Salleh, Albert Y.Zomaya
• Computational Methods for Fluid Dyanamics by Joel
  H. Ferziger. Milovan Peric
• J. J. MOR, A collection of nonlinear model
  problems, in Computational Solutions of Nonlinear
• Systems of Equations, E. L. Allgower and K.
  Georg, eds., Lectures in Applied Mathematics,
• American Mathematical Society, Providence, RI
• Y. SAAD, Krylov subspace methods for solving
  large unsymmetric linear systems, Math.
  Computation

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QUESTIONS