Error Analysis of Numerical Methods
Error Analysis of Numerical Methods#
In order to identify the essential properties of numerical methods, we define basic notions 1.
For the given ODE system
we define \(y(t_n)\) as the solution of IVP evaluated at \(t=t_n\), and \(y_n\) is a numerical approximation of \(y(t_n)\) at the same location by a generic explicit numerical scheme (no matter explicit, implicit or multi-step scheme):
where \(h\) is the discretization step for \(t\), i.e., \(h=t_{n+1}-t_n\), and \(\phi(t_n,y_n,h)\) is the increment function. We say that the defined numerical scheme is consistent if \(\lim_{h\to0} \phi(t,y,h) = \phi(t,y,0) = f(t,y)\).
Then, the approximation error is defined as
The absolute error is defined as
The relative error is defined as
The exact differential operator is defined as
The approximate differential operator is defined as
Finally, the local truncation error (LTE) is defined as
In practice, the evaluation of the exact solution for different \(t\) around \(t_n\) (required by \(L_a\)) is performed using a Taylor series expansion.
Finally, we can state that a scheme is \(p\)-th order accurate by examining its LTE and observing its leading term
where \(C\) is a constant, independent of \(h\), and \(H.O.T.\) are the higher order terms of the LTE.
Example: LTE for Euler’s scheme
Consider the IVP defined by \(y' = \lambda y\), with initial condition \(y(0)=1\).
The approximation operator for Euler’s scheme is
then the LTE can be computed by
where we assume \(y_n = y(t_n)\).