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:
“The poet’s eye, in a fine frenzy rolling,
Doth glance from heaven to earth, from earth to heaven;
And as imagination bodies forth
The forms of things unknown, the poet’s pen
Turns them to shapes, and gives to airy nothing
A local habitation and a name.”
—William Shakespeare (1564–1616)
“Knowledge, like matter, [my father] would affirm, was divisible in infinitum;Mthat the grains and scruples were as much a part of it, as the gravitation of the whole world.—In a word, he would say, error was error,—no matter where it fell,—whether in a fraction,—or a pound,—’twas alike fatal to truth.”
—Laurence Sterne (1713–1768)