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
Read more about this topic: Euler Method
Famous quotes containing the words local and/or error:
“America is the world’s living myth. There’s no sense of wrong when you kill an American or blame America for some local disaster. This is our function, to be character types, to embody recurring themes that people can use to comfort themselves, justify themselves and so on. We’re here to accommodate. Whatever people need, we provide. A myth is a useful thing.”
—Don Delillo (b. 1926)
“They have their belief, these poor Tibet people, that Providence sends down always an Incarnation of Himself into every generation. At bottom some belief in a kind of pope! At bottom still better, a belief that there is a Greatest Man; that he is discoverable; that, once discovered, we ought to treat him with an obedience which knows no bounds. This is the truth of Grand Lamaism; the “discoverability” is the only error here.”
—Thomas Carlyle (1795–1881)