# 2.4 Analysis of Finite Difference Methods

## 2.4.4 Finite Difference Methods in Matrix Form

Consider the one-dimensional convection-diffusion equation,

 $$\displaystyle \dfrac {\partial U}{\partial t} + u \dfrac {\partial U}{\partial x} -\mu \dfrac {\partial ^2 U}{\partial x^2}$$ $$\displaystyle =$$ $$\displaystyle 0.$$ (2.73)

Approximating the spatial derivative using the central difference operators gives the following approximation at node $$i$$,

 $$\displaystyle \dfrac {d U_ i}{d t} + u_ i \delta _{2x} U_ i - \mu \delta _ x^2 U_ i$$ $$\displaystyle =$$ $$\displaystyle 0$$ (2.74)

Assembling all of the nodal states into a single vector

 $$\displaystyle U = \left( U_0,\, U_1,\, \ldots ,\, U_{i-1},\, U_ i,\, U_{i+1},\, \ldots ,\, U_{N_ x-1},\, U_{N_ x}\right)^ T,$$ (2.75)

gives a system of coupled ODEs of the form:

 $$\displaystyle \dfrac {d U}{d t}$$ $$\displaystyle =$$ $$\displaystyle A U + b \label{equ:linSys}$$ (2.76)

where $$b$$ will contain the boundary condition related data. The matrix $$A$$ has the form:

 $$\displaystyle A$$ $$\displaystyle =$$ $$\displaystyle \left[\begin{array}{cccc} A_{0,0} & A_{0,1} & \ldots & A_{0,N_ x} \\ A_{1,0} & A_{1,1} & \ldots & A_{1,N_ x} \\ \vdots & \vdots & \ddots & \vdots \\ A_{N_ x,0} & A_{N_ x,1} & \ldots & A_{N_ x,N_ x} \end{array} \right].$$ (2.77)

Note that row $$i$$ of this matrix contains the coefficients of the nodal values for the ODE governing node $$i$$. Except for rows requiring boundary condition data, the values of $$A_{i,j}$$ are related to the coefficients $$\alpha _ j$$ of the finite difference approximation. Specifically, for our central difference approximation

 $$\displaystyle A_{i,i-1}$$ $$\displaystyle =$$ $$\displaystyle \frac{u_ i}{2 \Delta x} + \frac{\mu }{\Delta x^2},$$ (2.78) $$\displaystyle A_{i,i}$$ $$\displaystyle =$$ $$\displaystyle - 2\frac{\mu }{\Delta x^2},$$ (2.79) $$\displaystyle A_{i,i+1}$$ $$\displaystyle =$$ $$\displaystyle -\frac{u_ i}{2 \Delta x} + \frac{\mu }{\Delta x^2},$$ (2.80)

and all other entries of row $$i$$ are zero. In general, the number of non-zero entries of row $$i$$ will correspond to the size of the stencil of the finite difference approximations used. We refer to Equation 2.76 as being semi-discrete, since we have discretized the PDE in space but not in time. To make this a fully discrete approximation, we can apply any of the ODE integration methods that we discussed previously.

Implementation Notes When explicit methods are used to represent a FD method, then the $$A$$ matrix does not need to be directly formed. Implementing a finite difference approximation at each node will suffice as we will only be interested in matrix-vector multiplication, i.e. we only need to apply $$A$$.

On the other hand, implicit methods require the solution of a linear system. The solutions methods often require the representation of $$A$$ as a matrix, and thus it must be constructed.