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
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.