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2. Introduction to Numerical Methods
2.5.2 Stability
A numerical solution method is said to be stable if it does not magnify the
errors that appear in the course of numerical solution process. For temporal
problems, stability guarantees that the method produces a bounded solution
whenever the solution of the exact equation is bounded. For iterative methods, a stable method is one that does not diverge. Stability can be difficult
to investigate, especially when boundary conditions and non-linearities are
present. For this reason, it is common to investigate the stability of a method
for linear problems with constant coefficients without boundary conditions.
Experience shows that the results obtained in this way can often be applied
to more complex problems but there are notable exceptions.
The most widely used approach to studying stability of numerical schemes
is the von Neumann's method. We shall describe it briefly for one scheme in
Chap. 6. Most of the schemes to be described in this book have been analyzed for stability and we shall state the important result when describing
each scheme. However, when solving complicated, non-linear and coupled
equations with complicated boundary conditions, there are few stability results so we may have to rely on experience and intuition. Many solution
schemes require that the time step be smaller than a certain limit or that
under-relaxation be used. We shall discuss these issues and give guidelines for
selecting time step size and values of under-relaxation parameters in Chaps.
6 and 7.
2.5.3 Convergence
A numerical method is said to be convergent if the solution of the discretized
equations tends to the exact solution of the differential equation as the grid
spacing tends to zero. For linear initial value problems, the Lax equivalence
theorem (Richtmyer and Morton, 1967) states that "given a properly posed
linear initial value problem and a finite difference approximation to it that
satisfies the consistency condition, stability is the necessary and sufficient
condition for convergence". Obviously, a consistent scheme is useless unless
the solution method converges.
For non-linear problems which are strongly influenced by boundary conditions, the stability and convergence of a method are difficult to demonstrate.
Therefore convergence is usually checked using numerical experiments, i.e. repeating the calculation on a series of successively refined grids. If the method
is stable and if all approximations used in the discretization process are consistent, we usually find that the solution does converge to a grid-independent
solution. For sufficiently small grid sizes, the rate of convergence is governed
by the order of principal truncation error component. This allows us to estimate the error in the solution. We shall describe this in detail in Chaps. 3
and 5.
2. Introduction to Numerical Methods
2.5.2 Stability
A numerical solution method is said to be stable if it does not magnify the
errors that appear in the course of numerical solution process. For temporal
problems, stability guarantees that the method produces a bounded solution
whenever the solution of the exact equation is bounded. For iterative methods, a stable method is one that does not diverge. Stability can be difficult
to investigate, especially when boundary conditions and non-linearities are
present. For this reason, it is common to investigate the stability of a method
for linear problems with constant coefficients without boundary conditions.
Experience shows that the results obtained in this way can often be applied
to more complex problems but there are notable exceptions.
The most widely used approach to studying stability of numerical schemes
is the von Neumann's method. We shall describe it briefly for one scheme in
Chap. 6. Most of the schemes to be described in this book have been analyzed for stability and we shall state the important result when describing
each scheme. However, when solving complicated, non-linear and coupled
equations with complicated boundary conditions, there are few stability results so we may have to rely on experience and intuition. Many solution
schemes require that the time step be smaller than a certain limit or that
under-relaxation be used. We shall discuss these issues and give guidelines for
selecting time step size and values of under-relaxation parameters in Chaps.
6 and 7.
2.5.3 Convergence
A numerical method is said to be convergent if the solution of the discretized
equations tends to the exact solution of the differential equation as the grid
spacing tends to zero. For linear initial value problems, the Lax equivalence
theorem (Richtmyer and Morton, 1967) states that "given a properly posed
linear initial value problem and a finite difference approximation to it that
satisfies the consistency condition, stability is the necessary and sufficient
condition for convergence". Obviously, a consistent scheme is useless unless
the solution method converges.
For non-linear problems which are strongly influenced by boundary conditions, the stability and convergence of a method are difficult to demonstrate.
Therefore convergence is usually checked using numerical experiments, i.e. repeating the calculation on a series of successively refined grids. If the method
is stable and if all approximations used in the discretization process are consistent, we usually find that the solution does converge to a grid-independent
solution. For sufficiently small grid sizes, the rate of convergence is governed
by the order of principal truncation error component. This allows us to estimate the error in the solution. We shall describe this in detail in Chaps. 3
and 5.