Exercise 2: Finite Difference Method
Date of exercise: 15th May 2019
Last update: 20th May 2019 by Lukas Babor (Lukas.Babor@tuwien.ac.at)
2.1 Derivation of finite difference scheme from Taylor expansion
Consider a continuous function 
and a grid of N discrete points 
with uniform spacing 
, such that
and a grid of N discrete points with uniform spacing , such that
where 
and 
are the boundaries of a domain 
. "Discretization" of the function on the grid defines a column vector 
containig the values of the function evaluated at the points , i.e.:
and are the boundaries of a domain . "Discretization" of the function on the grid defines a column vector containig the values of the function evaluated at the points , i.e.:
With finite difference method we approximate the derivative of the function at some point using the values of the function evaluated at grid points in the neighbourhood of the point . For example, the second-order centered finite-difference approximation to a first derivative is
at some point using the values of the function evaluated at grid points in the neighbourhood of the point . For example, the second-order centered finite-difference approximation to a first derivative is 
Finite-difference schemes can be derived from Taylor expansion of the function u around the point . This also provides the expresion for the error term 
where n is the order of accuracy of the scheme. The derivation of a finite-difference scheme procedes as follows:
. This also provides the expresion for the error term where n is the order of accuracy of the scheme. The derivation of a finite-difference scheme procedes as follows:We start by assuming a certain form of the scheme, where we select which points we want to use. Taking the example above we have
Next we expand the values of the function 
with Taylor series from the point :
with Taylor series from the point :
and substitute the expansions into the expected form
Now we have to form a system of two algebraic equations to determine the coefficients a and b. This can be done by matching the derivatives on the left and right hand side of equation (4):
which leads to 
. Substituting 
into (2) provides the finite difference scheme (1). We still have to determine the order of accuracy through the error term ε. For that we substitute a and b into (4):
. Substituting into (2) provides the finite difference scheme (1). We still have to determine the order of accuracy through the error term ε. For that we substitute a and b into (4):
thus
which has two important consequences. Firstly, the scheme is second-order accurate since the leading error term is proportional to 
. Secondly, the leading error term does not depend on 
, so it does not contribute to artificial diffusivity.
. Secondly, the leading error term does not depend on , so it does not contribute to artificial diffusivity.In the same way one can derive finite-difference schemes for a non-uniform grid spacing 
. Although the schemes for non-uniform grids have often lower order of accuracy, they allow localised refinement of the function which can be a significant advantage. Using sensible non-uniform grids is therefore generally more efficient.
. Although the schemes for non-uniform grids have often lower order of accuracy, they allow localised refinement of the function which can be a significant advantage. Using sensible non-uniform grids is therefore generally more efficient.2.2 Matrix representation of finite difference schemes
The finite difference scheme (1) defines a vector of approximate first derivatives of at grid points . For simplicity we will use the notation
at grid points . For simplicity we will use the notation
Our aim is to represent the finite difference derivative 
with a discretization matrix 
such that
with a discretization matrix such that
where the superscript 1 denotes the first derivative, while the subscript x denotes the direction of the derivative (for the case of partial derivatives). Compilation of the matrix from (1) at interior grid nodes 
is straightforward. Its only non-zero entries are the first lower and upper diagonal corresponding to the coefficients 
and 
respectively. The centered scheme is, however, not defined for boundary nodes 
(the first and the last row of the matrix). We can approximate the derivatives on boundaries by one-sided differences, e.g.:
is straightforward. Its only non-zero entries are the first lower and upper diagonal corresponding to the coefficients and respectively. The centered scheme is, however, not defined for boundary nodes (the first and the last row of the matrix). We can approximate the derivatives on boundaries by one-sided differences, e.g.:
The discretization matrix then reads
In MatLab we can construct such matrices with the command spdiags
N = 4; % Number of grid points
dx = 1/(N-1); % Grid spacing
a = -1/(2*dx) ; % Coefficients of FD scheme
b = 1/(2*dx);
e = ones(N,1); % Column vector of ones
D_x = spdiags([a*e b*e], [-1 1], N,N ); % Matrix with D_x(i,i-1)=a; D_x(i,i+1)=b
D_x(1,: ) = 0; % Replaces first row of the matrix
D_x(1,1 ) = -1/dx;
D_x(1,2 ) = 1/dx;
D_x(N,: ) = 0; % Replaces last row of the matrix
D_x(N,N-1) = -1/dx;
D_x(N,N ) = 1/dx;
spy(D_x); % Plots the sparsity pattern of the matrix
