Numerically following the solution of an evolution equation in time is one of the central tasks of numerical analysis and has received over the years vast and repeated attention, see   . In this note we consider some essential and interesting, but possibly not widely known, aspects of the numerical solution of initial value problems. The rationale behind the procedures advocated here for the accurate and stable solution of the initial value problem, is often to greatly facilitate their introduction in the classroom. For the first order problem, we consider the usefulness of implicit methods, see  , for capturing multiple solutions emanating from a point of bifurcations, and also simple procedures for determining the accuracy and stability of multistep methods. For the second order problem, see  , we consider the means of slowing and advancing the computed speed of the numerical solution of the equation of motion.
2. The Generalized Mean Value Theorem—Taylor’s Theorem
The decisive significance of Taylor’s theorem (as can be looked up in any elementary calculus textbook) to applied mathematics in general, and to numerical analysis in particular, is that it ascertains that every differential function looks locally like a polynomial. Polynomials having the advantage of being easily computed, differentiated and integrated, relieving us by their use of the burden of possibly high complications in the symbolic manipulation of functions, often only implicitly given as the solution of an initial value problem (IVP) or a boundary value problem (BVP). We look at this theorem here from an unusual angle.
The mean value theorem (MVT) states that if function f(x), , , is continuous in the closed interval [0, x] and differentiable on the open interval (0, x), then point ξ exists, strictly inside the interval, , at which the slope of the chord equals to the slope of the tangent line to f(x) at point ξ, or
implying that f(x) is nearly linear near x = 0.
The MVT theorem is a direct result, of the geometrically intuitively plausible, Rolle’s theorem. The generalized mean value theorem (GMVT) is a result of the application of the MVT (or Rolle’s theorem) to the higher order derivative functions of f(x). Here it is in its most concise form: Let function f(x) be continuous at x = 0, and such that
Then, function f(x) may be expressed in the form
implying that if f(n)(x) is bounded near x = 0, then f(x), like xn, is small if .
3. Where Is ξ
We consider first the simplest case of n = 1, for which Equation (3) is
such that .
We further assume that, approximately
and have from this that
Annulling the first two terms of the above equation results in
or generally, for any n in Equation (2)
4. Polynomial Approximations
Some examples will convince us of the decisive usefulness of the GMVT theorem to numerical analysis. As a first example, we use the theorem to get a good polynomial approximation to ex near x = 0. We start by writing
and propose to fix free parameters a and b such that , , namely, such that
Now, by fundamental Equation (3), we may write r(x) of , , as
providing us with a good linear polynomial approximation to ex in the vicinity of x = 0.
Moreover, since ex is an increasing function, we readily obtain from Equation (12) the strict inequalities
5. Improving the Polynomial Approximation with ξ
The reason the lower bound on ex in the above inequality (14) is better then the upper bound is due to the fact that as the order of the approximation increases, ξ moves rather ever closer to the osculation point x = 0. Here, since , , , then, according to Equation (9), ξ = x/3, nearly, if .
Replacing eξ in Equation (12) by 1 + ξ, with ξ = x/3, we obtain, forthwith, the better approximation
Otherwise we may start from
take , expand further
To have in the above equation as many correct terms as possible we set
6. Trigonometric Function Approximations
Taylor’s theorem is not restricted to polynomials, but may be advantageously used to directly construct other approximating functions. For instance, we may start with
and use free numbers a and b to enforce and , to have
Consequently, the Taylor, or the GMVT, form of r(x) is
or, asymptotically, as ,
7. Rational Function Approximations
Rational approximations are also desirable, and efficient. Here we start with, say
of the three free parameters a, b, c. Imposing on r(x) the conditions
we readily have
If the point of osculation is not x = 0, but x = a, then x in the theorem is shifted to x − a.
8. First Order Linear Homogeneous Recursions
In the numerical integration of IVPs we are constantly confronted by the need to solve linear homogeneous recursions.
The first order, homogeneous, recursion
where b is a constants independent of n, is brought, without much ado, to the explicit, closed-form, representation
by merely repeating the recursion. We note that if , then yn keeps growing with n, while if , then , as .
9. Second Order Linear Homogeneous Recursions
Next we consider the three-term homogeneous recursion.
Let the sequence be generated by the homogeneous recursion
with coefficients b and c assumed independent of n. Recursion (30) is satisfied by provided z is a root the characteristic equation
In case the two roots z1, z2 of Equation (31) are distinct, , then by the linearity of the recursion we have the general solution of this recursion in the form
with c1 and c2 determined by the initial conditions y0 and y1.
In case the roots of Equation (31) are equal, , , then we verify that , with
In case the roots of Equation (31) are complex conjugates, , , , then we may put z in the form
where . Now, becomes
with c1,c2 fixed by the initial conditions y0, y1.
For example, the recursion
10. The Advantage of an Implicit Method at a Branching Point
Implicit methods for the numerical integration of the first-order IVP hold some stability advantages, but they may require the solution of a nonlinear equation for the next predicted value.
At a point of bifurcation they hold the extra advantage of capturing multiple solutions, otherwise missed by an explicit method. Here is an example. The initial value problem
is solved by both
Using the Euler explicit method
where y1 is an approximation to y(τ), we have
which is only the first solution y(t) = 1 of IVP (38).
Using the implicit method
then the quadratic equation
for y1, solved by
a first , and a second (45)
as compared with
11. Sensitivity to Initial Conditions
The solution of the IVP
and if , then , but if , then as .
12. Determination of the Order of Consistency of Multistep Methods
To fully, yet concisely, demonstrate the consistency and stability issues in the integration of first order IVP, and their resolution, we shall look in detail at the general two-step method
in which y2 is the computed approximation to the correct y(2τ), in which err is the error y(2τ) − y2, and in which α0, α1, β0, β1 are free parameters to be determined for highest accuracy and method stability.
In accordance with Taylor’s theorem we require, for the highest possible order of consistency, that the calculated y2 is the correct y(2τ), namely err = 0, for
leading to the system of equations
and then to
in which we leave α0 free for now, to use it next to guarantee the stability of the method.
According to Taylor’s theorem the worst case error arises from the next order polynomial, or the function with a constant third derivative. Accordingly, we take next
and have from Equation (49) and Equation (51) that
which is the local error per step. After n steps the error rises to the global O(τ2).
13. Stability of the Multistep Method
The following Lemma is greatly useful for ascertaining the stability of an integration method.
Lemma. Let real roots z(τ) of the characteristic equation for the integration scheme of the first-order IVP be such that . Then, a sufficient condition for the (conditional) stability of the method is that at , and that at , .
Proof. It results directly from the continuity and differentiability of z(τ) that if and at τ = 0, then z(τ) < 1 for some τ > 0. See also  .
Specifically, for the model IVP
the two-step method of Equation (49) becomes
of the characteristic equation
At τ = 0 the equation reduces to
which is of the two roots
To verify the stability of the method we seek at τ = 0.
Implicitly differentiating characteristic Equation (57) with respect to τ we have
At τ = 0, the above equation reduces to
and for we obtain from the above equation that
and since for stability z1(τ) needs to come down at τ = 0, hence .
14. The Adam-Bashforth Method
In this method , for which the characteristic equation reduces to
We set z = 1 in the above equation and get from it the corresponding τ = 0. Then we set in the above equation z = −1 and obtain from it τ = 1, at which , implying that the method is stable for .
15. Other Possibilities
For the characteristic equation of the multistep method becomes
and by implicit differentiation with respect to τ
where . At τ = 0, z = 1, the above equation yields , and the method is stable for some τ. We set z = 1 into the characteristic equation and get for it but τ = 0. We set into the characteristic equation z = −1 and get from it τ = 7/4, implying that the method is stable now for .
16. Amplification of an Initial Error
We shall consider now the application of the two-step method of the previous section to the IVP
For , , and with , the two-step method becomes
for which the characteristic equation is
Say we start the method with , , such that
resulting in , , and
17. A Three Step Method
The integration method
is correct for y = 1, y = t, y = t2 and y = t3. For
we have from Equation (75) that
For , the characteristic equation of the three step method becomes
and at τ = 0 it reduces to
, of roots . (79)
Implicit differentiation of Equation (78) with respect to τ yields
and at τ = 0, z = 1, we have that , so that near τ = 0
and the method is stable.
We further have from Equation (79) that at τ = 6/11, z3 = −1 and z1 = z2 = 1/2.
At the repeating root z = 0, the derivative function does not exist. Instead we write τ in terms of z as
and if z = 0, nearly, then
18. Advancing and Retarding the Computed Motion
Next, we turn our attention to the second order IVP, see  and  , as typified by the model second order problem
which we propose to approximate as
The characteristic equation of this method is
For a sufficiently small τ, the roots of the characteristic equation are complex and . Hence the closed-form prediction of the computed y at step n
where c1 and c2 are determined by the initial conditions.
From the characteristic Equation (87) we have that
To drop the τ2 term in the above equation we take α = 0, and are left with
suggesting that θ may be advanced or retarded relative to τ with a proper choice of β. For instance, for β = −1/12 we have that θ = τ, nearly.
19. Integration of the Equation of Motion Linear Prediction
Inasmuch as the initial conditions to the second order equation of motion are usually given in terms of initial position and initial velocity, we prefer to reduce the second order problem into a coupled system of first order equations for position and velocity, and follow them in tandem.
To remain both concise and specific, we consider the model initial value problem
where x = x(t), y = y(t), t > 0, and where denotes differentiation with respect to time t. This initial value problem, that coincides with a single second order problem, is solved by , representing a constant circular motion of period .
We propose to follow the initial value problem with the explicit scheme
in which τ is the time step, where x1 = x(τ) and y1 = y(τ), approximately, and with the coefficients α0, α1 to be presently determined by stability and accuracy considerations.
With , , system (93) becomes the system of recursions
that explicitly produces x1 and y1 out of x0 and y0, and then x2 and y2 out of x1 and y1, and so on up to xn and yn. System (94) is solved by , for magnification factor z that satisfies the pair of linear equations
for any x0 and y0. Equation (95) is recast in the matrix-vector form to assume the form
and the condition that it have a nontrivial solution is that the determinant of the system’s matrix of coefficients be zero, leading to the characteristic equation
for z. The periodic nature of the solution to this initial value problems dictates that z be complex. Let be the modulus of complex z. If , then as , and if , then as . To avoid these undesirable eventualities of an artificial energy sink and an artificial, numerically induced, energy source, we select α0 = 0 in Equation (94), and are left with the reduced characteristic equation
that possesses two complex roots z1 and z2 such that .
where i2 = −1, and z is complex if
By the fact that , the complex solution to Equation (98) is energy conserving, and may be written as
Now, xn and yn are generally written as
with constants determined by the initial conditions. Given x0 = 1, y0 = 0 we get from Equation (94), x1 = 1, y1 = α1τ. Writing xn and yn in Equation (102) for n = 0 and n = 1 we obtain the two systems of linear equations
readily solved for as
in which , , . Writing z1 and z2 in terms of θ recasts Equation (103) into the form
and we have from Equation (105) that
with which we finally get
as the general numerical solution to our initial value problem.
20. Period Control
A cycle is completed when or . Then, according to Equation (108) yn = 0 and xn = 1. From and T = nτ we obtain the computed period as
and to retain we select α1 in Equation (93) so as to guarantee τ = θ or . This condition becomes, in view of Equation (102),
leading to the quadratic equation
for α1, and resulting in
if τ is small.
21. Quadratic Prediction
Inclusion of the acceleration in the prediction of x1 suggests the higher order scheme
that becomes for
Substitution of x1 = zx0, y1 = zy0 in Equation (115) results in the system
from which we obtain the quadratic characteristic equation
for magnification factor z. To assure for the complex roots of Equation (117) we set α0 = 1 and are left with
where β = 1 + α1. The two roots of Equation (118) are
and z is complex if
Because we may write the complex roots of Equation (118) as
The numerical scheme is period conserving if τ = θ, or . This is assured, according to Equation (121), by β such that
if τ is small.
We accomplished here showing how to routinely determine the consistency and stability of any multistep method, explicit as well as implicit, for the stepwise integration of the first order initial value problem. We have also demonstrated here the advantage of using implicit methods to capture different solutions emanating from a branch-off point. For the integration of the second order equation of motion we have shown how to advance and retard the motion of the computed solution.