- The quadrature of all figures follow from the inverse method of tangents, and thus the whole science of sums and quadratures can be reduced to analysis, a thing nobody even had any hopes of before. (Leibniz)
- Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method. (Leibniz
Let us now study the effect of the time step in solution of the basic IVP
by Forward Euler time stepping
We compare taking one step with time step with two steps of time step , for a given :
where is computed with time step , and we assume that the same intial value for is used so that . Assuming that is Lipschitz continuous with Lipschitz constant , we then find that
Summing now the contributions from all time steps with , where is a final time, we get using that ,
where thus is computed with time step and with time step .
Repeating the argument with successively refined times step , we get
for the difference between computed with time step and computes with vanishingly small time step, since
We have now proved the Fundamental Theorem of Calculus:
Theorem If is Lipschitz continuous, then the function defined by Forward Euler time-stepping with vanishing time step, solves the IVP: for , .
Understanding the Fundamental Theorem
The proof shows what it means to understand the Fundamental Theorem of Calculus: This is to realize that (letting denote a finite time step and a vanishingly small step)
as a consequence of
where the sum is referred to as a Riemann sum, with the following bound for the difference
assuming is Lipschitz continuous with Lipschitz constant .
In other words, understanding the integral of a function , means to understand that:
- is determined by Riemann sums with vanishingly small step size, as the solution to the IVP , ,
- the difference between two Riemann sums with mesh size and , is bounded by (or more precisely by ).
Even Better Understanding
As a serious student, you now probably ask: In precisely what sense the differential equation is satisfied by an Euler Forward solution with time step ? It certainly is so constructed, but can we get a direct verification? One way to do this is to associate a continuous piecewise linear function determined by the values at the discrete time levels ,again denoted by . We then have on each interval , by the definition of :
from which we conclude that
We can thus say that satisfies the differential equation for all with a precision of . In other words, the residual is smaller than .
We have now understood the Fundamental Theorem even better, right?
We shall see below that extending a function defined on a discrete set of points to a continuous piecewise linear function, is a central aspect of approximation in general and of the Finite Element Method FEM in particular.
To Think About
- What is fundamental about the Fundamental Theorem?
- Why is ? (compare with last argument)
- What is the effect of finite precision computation according to Constructive Calculus in Finite Precision? (see below)
- What is the Riemann sum error using the Trapezoidal Rule ?
Fundamental Theorem in Finite Precision
Lipschitz continuity in the presence of finite precision can be defined as follows: A real-valued function of a real variable is Lipschitz continuous with Lipschitz constant in finite precision , if for all and
We see that here will effectively be bounded below by . With this extension of the concept of Lipschitz continuity to finite precision, the first step of the above proof takes the form
In the second step, the repetition with successively refined times step , is performed until for some natural number , which gives
for the difference between computed with time step and computed with time step . Here is the 2-logarithm of and thus is a constant of moderate size (not large). In essence, we thus obtain the previous estimate with replaced by and appears as a lower bound of the time step