Truncation errors in numerical integration are of two kinds:
Contents
- Definitions
- Local truncation error
- Global truncation error
- Relationship between local and global truncation errors
- Extension to linear multistep methods
- References
Definitions
Suppose we have a continuous differential equation
and we wish to compute an approximation
Suppose we compute the sequence
The function
Local truncation error
The local truncation error
More formally, the local truncation error,
The numerical method is consistent if the local truncation error is
Furthermore, we say that the numerical method has order
Global truncation error
The global truncation error is the accumulation of the local truncation error over all of the iterations, assuming perfect knowledge of the true solution at the initial time step.
More formally, the global truncation error,
The numerical method is convergent if global truncation error goes to zero as the step size goes to zero; in other words, the numerical solution converges to the exact solution:
Relationship between local and global truncation errors
Sometimes it is possible to calculate an upper bound on the global truncation error, if we already know the local truncation error. This requires our increment function be sufficiently well-behaved.
The global truncation error satisfies the recurrence relation:
This follows immediately from the definitions. Now assume that the increment function is Lipschitz continuous in the second argument, that is, there exists a constant
Then the global error satisfies the bound
It follows from the above bound for the global error that if the function
Extension to linear multistep methods
Now consider a linear multistep method, given by the formula
Thus, the next value for the numerical solution is computed according to
The next iterate of a linear multistep method depends on the previous s iterates. Thus, in the definition for the local truncation error, it is now assumed that the previous s iterates all correspond to the exact solution:
Again, the method is consistent if
The relation between local and global truncation errors is slightly different from in the simpler setting of one-step methods. For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors. Linear multistep methods that satisfy the condition of zero-stability have the same relation between local and global errors as one-step methods. In other words, if a linear multistep method is zero-stable and consistent, then it converges. And if a linear multistep method is zero-stable and has local error