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5.7 Convergence and Stability 165

5.7 Convergence and Stability

If U denotes the exact solution of a partial-differential equation with indepen-
dent variables x and £, and the exact solution of the difference equations is
u, then the difference (U — u) is called the discretization error. Its magnitude
at any grid point depends on the mesh widths and on the number of finite-
differences in the truncated series used to approximate the derivatives. The
finite-difference approximations are said to be converged when the discretiza-
tion error approaches to zero as mesh widths At and Ax both approach to
zero.
The round-off error refers to the difference between the exact solution of
the difference equations, u and its approximating difference equation, N. It
will be zero if in the solution of the finite-difference equations the calculations
can be performed for an infinite number of decimal places. In practice this is
not possible and the calculations are performed for a finite-number of decimal
places, leading to round-off errors. In general, the solution of the finite-difference
equations is stable when the cumulative effect at all the rounding errors is
negligible.
More specifically, if errors e™ are introduced at the grid points (i,n), and
each value of \ef\ is less than 6, then the difference equations are stable when the
maximum value of the round-off error (u — N) approaches zero as 6 approaches
zero and does not increase exponentially with the number of columns or rows
in the calculation, i.e., with 1 or n. The latter condition is necessary because in
certain cases the errors may not decrease exponentially with i or n but persist
as linear combinations of the initial errors. In many cases they are numerically
tolerable provided their sum remains much smaller than u.
The growth of errors in the computations can be examined by expressing the
equations in matrix form and analyzing the eigenvalues of an associated matrix
or by expressing them in a finite Fourier series. Here, due to its simplicity, the
Fourier series method, which is also known as the von Neumann analysis, is
discussed. For simplicity, the discussion is restricted to a single equation. The
stability analysis for systems of equations can be found in several references,
see for example, Hirsch [1].
To describe the stability analysis of a numerical scheme for a linear equation
with the von Neumann analysis, consider the explicit approximation to the
unsteady heat conduction equation, Eq. (4.4.3a). If T/ 1 is the exact solution of
the difference equation and T™ the actual computed solution, then e™ in

T? = f? + e? (5.7.1)

represents the error at time step n at grid point i. Substituting Eq. (5.7.1) into
the difference equation, Eq. (4.4.3a),