Local Truncation Error
The local truncation error of the Euler method is error made in a single step. It is the difference between the numerical solution after one step, and the exact solution at time . The numerical solution is given by
For the exact solution, we use the Taylor expansion mentioned in the section Derivation above:
The local truncation error (LTE) introduced by the Euler method is given by the difference between these equations:
This result is valid if has a bounded third derivative.
This shows that for small, the local truncation error is approximately proportional to . This makes the Euler method less accurate (for small ) than other higher-order techniques such as Runge-Kutta methods and linear multistep methods, for which the local truncation error is proportial to a higher power of the step size.
A slightly different formulation for the local truncation error can be obtained by using the Lagrange form for the remainder term in Taylor's theorem. If has a continuous second derivative, then there exists a such that
In the above expressions for the error, the second derivative of the unknown exact solution can be replaced by an expression involving the right-hand side of the differential equation. Indeed, it follows from the equation that
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