Finite difference method - Mahidol University · 2020. 8. 27. · derivatives with finite...

transcript

Finite Difference Method

Chaiwoot Boonyasiriwat

August 27, 2020

▪ The value of a derivative of function can be

approximated by finite difference formulas based on

function values at discrete points.

▪ Differential equations can then be solved by replacing

derivatives with finite difference approximations.

▪ This discretization is called finite difference method.

▪ This leads to a system of algebraic equations which can

be solved using numerical methods on a computer.

▪ A numerical solution from FDM are only known at

discrete points in space and/or time.

▪ Accuracy of the solution depends on the order of

accuracy of the finite difference approximations used.

Introduction to FDM

▪ Let u(x) be a function of one variable.

▪ A derivative of u(x) can be approximated based on the

values of the function at discrete points.

▪ Recall the definition of derivative

▪ If the limit is dropped out, we then obtain a formula for

approximating the first derivative

▪ This formula is a one-sided finite difference

approximation.

Finite Difference Approximation

▪ Forward finite-difference scheme

▪ Backward finite-difference scheme

▪ Centered finite-difference scheme

Commonly Used Formulas

Example 1

LeVeque (2007, p. 4)

D0u(x) is obviously more accurate than D-u(x) and D+u(x)

Another finite-difference scheme

Asymmetric Approximation

First-order accurate approximation

Second-order accurate approximation

Third-order accurate approximation

Order of Accuracy

Let u(x) = sin(x).

Approximating u'(x) at x = 1 using various values of

step length h, we obtain numerical errors from each

approximation compared to the exact solution:

Example 2

LeVeque (2007, p. 4-5)

According to the results, we can see that

In general, the approximation error can be written as

where p is the order of approximation.

Example 2 (continued)

Absolute errors versus h on a log-log scale

Taylor’s series expansion is the standard approach for

analyzing the truncation error of a finite-difference

approximation.

Truncation Error

Rearranging the first expansion yields

Truncation Error (continued)

Truncation error

Big O Notation: Let f and g be real-valued functions.

One can write

if and only if there exists a positive real number M and a

real number x0 such that

If a truncation error is

Big O Notation

https://en.wikipedia.org/wiki/Big_O_notation

To make finite difference expressions concise and simple

to handle, we usually use the following index notations.

Index Notation and Convention

▪ FD stencil is a graphical representation of a finite

difference approximation.

▪ Stencil makes a complicated FD expression looks

simpler.

FD Stencil

i - 2 i - 1 i i + 1 i + 2

-1/12 4/3 -5/2 4/3 -1/12

▪ Finite difference coefficients can be derived using the

method of undetermined coefficients.

▪ Example 3: Suppose we want to derive a one-sided FD

approximation to u'(x) based on the function values

u(x), u(x – h), and u(x – 2h)

First, write a formula as a linear combination of the

function values as

where a, b, and c are unknowns to be determined so that

we obtain the highest accuracy possible.

Deriving FD Coefficients

Rearranging the Taylor’s expansions of u(x – h), and u(x

– 2h), we obtain

To obtain an approximation of u'(x), we need

Use the method of undetermined coefficients to determine

the coefficients of a finite difference approximation to

u"(x) based on the function values u(x – h), u(x), and

u(x + h). What is the order of accuracy of the formula?

Exercise

The method of undetermined coefficients can be extended

to compute FD coefficients for using

points:

Assume that u(x) is at least n+1 times differentiable in the

interval containing and all stencil points.

Taylor series expansions of u at each point xi about are

LeVeque (2007, p. 10)

“We want to find a linear combination of these values that

agrees with as well as possible. So we want

where p is as large as possible.”

It can be shown that the coefficients ci satisfy

for i = 1, …, n. This leads to a linear system which has a

unique solution.

LeVeque (2007, p. 10)

% k = derivative order

% x = location of approximation

% xi = set of FD stencil points

function c = fdcoeff(k,x,xi)

n = length(xi);

A = ones(n);

xrow = (xi(:)-x)';

for i=2:n

A(i,:)=(xrow.^(i-1))./factorial(i-1);

b = zeros(n,1); b(k+1) = 1;

c = A\b; c = c(:);

MATLAB Code

LeVeque (2007, p. 11)

▪ “Another way to derive FD coefficients is to

approximate the function u(x) by some polynomial p(x)

and then use p'(x) as an approximation to u’(x).”

(LeVeque 2007, p. 8)

▪ If we want to use the value of function u(x) at n + 1

points, say, xi, i = 0, 1, …, n, p(x) will be a polynomial

of degree up to n:

where ai(x) are the Lagrange basis functions.

▪ The degree of polynomial is the order of approximation.

Using two points xi and xi+1, the interpolant is

and its derivative is

which is the forward FD formula.

Example 4

The interpolant can be written in terms of Lagrange

basis polynomial Li(x) as

Lagrange Interpolant

Determine the FD coefficients for u"(x) using 3 points

by differentiating the corresponding

polynomial interpolant.

Exercise

▪ Recall the centered FD approximation

▪ Let’s assume that the problem is periodic, i.e., and .

▪ For each index i we have one equation.▪ So we have totally n linear equations.

Differentiation Matrix

Trefethen (2000, p. 1)

▪ The linear system can be written in matrix form as

▪ This matrix is Toeplitz and circulant.

Image from Trefethen (2000, p. 2)

Differentiation matrix

▪ The linear system for a 4th-order FD approximation is

Example 4 (Trefethen, 2000, p. 3): Using 4th-order FD to approximate the first derivative of

using various values of N.

Order of Accuracy (Revisited)

( ) pE N DN −=

( ) pE h Ch=

h L N=

Using global interpolation (all points are used) will provide the highest-order approximation.

Spectral Method

Using the spectral method, the approximation will converge to exact solution very rapidly until rounding errors prevent any further improvement (Trefethen, 2000).

Spectral Method (continued)

Laplace’s equation:

Solutions of Laplace’s equation are analytic , i.e., the Cauchy-Riemann equations are satisfied, and called harmonic functions and sum of them is also a solution, i.e., the superposition principle.

Laplace and Poisson Equations

Reference: http://en.wikipedia.org/wiki/Laplace's_equation

Poisson’s equation

Consider 2D incompressible, irrotational flow

Show that the scalar velocity potential and the stream function defined by

satisfied the Laplace’s equation.Vector calculus identify:

Exercise: Fluid Flow

Reference: http://en.wikipedia.org/wiki/Laplace's_equation

Given the Maxwell’s equations

Show that the scalar potential defined by

satisfies the Poisson’s equation

Exercise: Electrostatics

Reference: http://en.wikipedia.org/wiki/Poisson's_equation

Given the Maxwell’s equations

Show that the vector potential A defined by(Coulomb gauge)

Exercise: Magnetostatics

Reference: http://farside.ph.utexas.edu/teaching/em/lectures/node40.html

Given the Newton’s law of gravitation

and Gauss’s law for gravity

Show that the gravity potential defined by

Exercise: Gravity

Reference: http://en.wikipedia.org/wiki/Gauss's_law_for_gravity

Given the 1D boundary value problem

Using the centered finite difference

we obtain the difference equation

1D Poisson Equation

Reference: LeVeque (2007, p. 15)

Dirichlet BC

This leads to a system of m linear equations

1D Poisson Equation

Reference: LeVeque (2007, p. 16)

▪ In this case, matrix A is tridiagonal. The tridiagonal linear system can be efficiently solved by the Thomas’ algorithm which is a special case of Gaussian elimination.

▪ A tridiagonal linear system can be written as

Thomas Algorithm

Image from http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

The solution is obtained by back substitution given by

Thomas Algorithm

Local truncation error can be determined by substituting the true solution into the difference equation.

Although we do not know the true solution, but if we assume a smooth solution, we can use the Taylor’s expansion to obtain

Local Truncation Error

LeVeque (2007, p. 17)

Using the Poisson equation we obtain

This leads to the linear system

where is the vector true solutionis the vector of truncation error

Local Truncation Error

LeVeque (2007, p. 17)

Global error defined by satisfies the boundary value problem

Since , the global error is roughly

Global Error

LeVeque (2007, p. 18)

To determine if the FD scheme is stable, consider the linear system with solution

with norm

Since so the FD scheme is stable ifis bounded, i.e.

Stability

LeVeque (2007, p. 18-19)

Triangular

inequality

Higher-Order Methods

▪ Higher-order finite-difference methods• Use longer FD stencils.

▪ Richardson extrapolation• Use low-order FD solutions on two grid

levels to obtain higher-order solution.▪ Deferred correction

• Use low-order FD solution to approximate the local truncation error and then solve for the global error to obtain more accurate solution.

Recall that

Let . We than have

If we also use another step size , we also have

Solving for yields r-order accurate value

Richardson Extrapolation

Heath (2002, p. 369)

Richardson Extrapolation

LeVeque (2007, p. 53-54)

Let’s use second-order FD. is true solution.Coarse grid solutionFine grid solutionWe then have

Show that the extrapolated value

is 4th-order accurate

Deferred Correction

LeVeque (2007, p. 53-54)

First solve to obtain 2nd-order accurate solution.The global error satisfieswhere

We can use the approximate solution to approximate and obtain an estimated global error E to improve the accuracy of u.In this case, we can use the Poisson equation to obtain

Fick’s first law: Mass diffusionConcentration goes from regions with high concentration to regions with low concentration

where J is diffusion fluxD is diffusion coefficient is substance concentration

Fick’s Laws of Diffusion

http://en.wikipedia.org/wiki/Fick's_laws_of_diffusion

A drag or viscosity force exerting on spherical objects with low Reynolds numbers (e.g. small in size) moving through a viscous fluid is given by

where is dynamic viscosityr is radius of spherical objectsv is flow velocity

Stokes’ Law of Drag Force

http://en.wikipedia.org/wiki/Stokes'_law

Using Fick’s laws and Stokes’ law, Einstein was able to unify the physical diffusion of Fourier and stochastic diffusion of Laplace through the theory of Brownian motion (Einstein, 1905).

Mean squared displacement and time relation:

Stokes-Einstein relation:

Stokes-Einstein Relation

http://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)

where kB is Boltzmann’s constant

T is temperature

A diffusion process is governed by the diffusion equation

In mass transport, the process is characterized by the diffusion coefficient.

In heat conduction, the process is characterized by the diffusion coefficient and thermal conductivity.

Diffusion Equation

http://en.wikipedia.org/wiki/Diffusion_equation

Assuming a steady state and constant diffusion coefficient, the diffusion equation becomes the Poisson’s equation

which is an elliptic PDE.We can use 2nd-order FD to solve the problem.

Steady-State Problems

http://en.wikipedia.org/wiki/Diffusion_equation

5-point stencil 9-point stencil-4

Both stencils are second-order accurate but the 9-point stencil has an advantage when it is used with the deferred correction method.

Accuracy of 2D Stencils

LeVeque (2007, p. 64-65)

Consider the 1D time-dependent diffusion equation

We can solve the equation using the explicit FD

Truncation error:

Exercise: Derive the stability condition

1D Diffusion Equation

LeVeque (2007, p. 182)

where D2 is the 5-point stencil.This leads to the linear systemwhich must be solved simultaneously to get the approximate solution u.

Truncation error:

Crank-Nicolson Method

LeVeque (2007, p. 182)

▪ Einstein, A., 1905, On the motion required by the molecular kinetic theory of heat, of particles suspended in fluids at rest, Annalen der Physik 17, 549-560.

▪ Heath, M.T., 2002, Scientific Computing: An Introductory Survey, McGrawHill.

▪ LeVeque, R. J., 2007, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM.

▪ Trefethen, L. N. 2000, Spectral Methods in MATLAB, SIAM.

References

References▪ Heath, M. T., 2002. Scientific Computing: An

Introductory Survey, Second Edition, McGraw-Hill.▪ LeVeque, R. J., 2007, Finite Difference Methods for

Ordinary and Partial Differential Equation, SIAM.▪ Trefethen, L. N., 2000, Spectral Methods in MATLAB,

SIAM.▪ Einstein, A., 1905, On the motion required by the

molecular kinetic theory of heat, of particles suspended in fluids at rest, Annalen der Physik 17, 549-560.

Finite difference method - Mahidol University · 2020. 8. 27. · derivatives with finite...

Documents