Section 2.3 Convergence, Consistency and Stability of Finite Difference Methods
Unlike the ODEs case, for PDEs there is no general result about convergence and consistency of fixed-stepsize numerical methods except in the linear case. Hence, in this section we focus on linear parabolic or hyperbolic PDEs, namely PDEs of the type
\begin{equation*}
u_t=Lu
\end{equation*}
where \(L\) is a differential operator with respect to space coordinates. Definition 2.3.1.
Consider a rectangle \(R=[0,T]\times[0,L]\) and an integer \(M>1\text{.}\) By a Finite Difference fixed-stepsize method on \(R\) with time stepsize \(h\) and space step-size \(k=L/M\) we mean a map
\begin{equation*}
\Phi_{h}:\bR^{M+1}\to\bR^M.
\end{equation*}
This map represents the advance of the numerical solution from time \(t\) to time \(t+h\text{.}\)Example 2.3.2.
\begin{equation*}
\Phi_h(t,u_1,u_2,\dots,u_M)
= (a(t+h),u_2-\nu(u_2-u_1),\dots,u_M-\nu(u_M-u_{M-1})),
\end{equation*}
where \(\nu=h/k\) and the function \(a(t)\) in the expression above is the extra boundary condition needed by the FTBS method, representing the values of the solution at \(x=0\text{.}\) Using this map, the FTBS method can be written as
\begin{equation*}
(u_{n+1,1},u_{n+1,2},\dots,u_{n+1,M}) = \Phi_h(t_n,u_{n,1},u_{n,2},\dots,u_{n,M}).
\end{equation*}
Notice that \(\Phi_h\) does not depend on time since \(L\) does not.Definition 2.3.3.
The Finite Difference method \(\Phi_h\) has order of accuracy\(\;p\) in the interval \([0,T]\) if, given a solution \(u\) of the linear PDE \(u_t=Lu\text{,}\) we have that
\begin{equation*}
\|u(t+h)-\Phi_h u(t)\|=O(h^{p+1})
\end{equation*}
for \(h\to0\) and any \(t\in[0,T]\text{.}\) \(\Phi_h\) is consistent if it has order of accuracy \(p>0\text{.}\)Example 2.3.4.
\begin{equation*}
\Phi_h(u_{m,n}) = u_{m,n} + \frac{h}{2k}\left(u_{m,n+1}-u_{m,n-1}\right)
\end{equation*}
with
\begin{equation*}
k = h^{p/2}.
\end{equation*}
Then \(\Phi_h\) has accuracy order \(p\) for each \(p\in[0,2]\)Definition 2.3.5.
The Finite Difference method \(\Phi_h\) is convergent in the interval \([0,T]\) if, given a solution \(u\) of the linear PDE \(u_t=Lu\) with initial condition \(u(0)=u_0\text{,}\) we have that
\begin{equation*}
\lim_{n\to\infty}\|u(t)-\Phi^n_{t/n}u_0\|=0.
\end{equation*}
Definition 2.3.6.
The Finite Difference method \(\Phi_h\) is stable in the interval \([0,T]\) if, for some constant \(C>0\text{,}\) we have that
\begin{equation*}
\sup_{nh\in[0,T]}\|\Phi^n_{h}\|\leq C
\end{equation*}