Solving System of ODEs using Midpoint Method

This code uses Midepoint Method to Solve Multi Ordinary Differential Equations (ODE). The code has been built in way you can edit it easily. The output chart presents the numerical solution for all the given differential equations.

Input Requirements:

The differential equation you intended to calculated.
Initial Conditions.
Initial and Final Integral Terms.
Number of Interval Divisions (Optional)

About the Method:

Illustration of the midpoint method assuming that yn equals the exact value y(tn) The midpoint method computes yn+1 so that the red chord is approximately parallel to the tangent line at the midpoint (the green line).

In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation,

y

The explicit midpoint method is given by the formula

y_{n+1} = y_n + hf\left(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n, y_n)\right), \qquad\qquad (1e)

the implicit midpoint method by

y_{n+1} = y_n + hf\left(t_n+\frac{h}{2},\frac12 (y_n+y_{n+1})\right), \qquad\qquad (1i)

for n = 0,1,2,… Here, h is the step size — a small positive number, tn = t0 + n.h and yn is the computed approximate value of y(tn) The explicit midpoint method is also known as the modified Euler method, the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator.

The name of the method comes from the fact that in the formula above the function f giving the slope of the solution is evaluated at t = tn + h/2 (note : t is (tn + tn+1)/2) which is the midpoint between tn at which the value of y(t) is known and tn+1 at which the value of y(t) needs to be found.

A geometric interpretation may give a better intuitive understanding of the method. In the basic Euler’s method, the tangent of the curve at (tn, yn) is computed using f(tn, yn). The next value $yn+1$ is found where the tangent intersects the vertical line t = tn+1. However, if the second derivative is only positive between tn and tn+1, or only negative (as in the diagram), the curve will increasingly veer away from the tangent, leading to larger errors as h increases. The diagram illustrates that the tangent at the midpoint (upper, green line segment) would most likely give a more accurate approximation of the curve in that interval. This tangent is estimated by using the original Euler’s method to estimate the value of y(t) at the midpoint, then computing the slope of the tangent with f(). Finally, the improved tangent is used to calculate the value of yn+1 from yn. This last step is represented by the red chord in the diagram. Note that the red chord is not exactly parallel to the green segment (the true tangent), due to the error in estimating the value of y(t) at the midpoint.

The local error at each step of the midpoint method is of order O(h^3), giving a global error of order O(h^2). Thus, while more computationally intensive than Euler’s method, the midpoint method’s error generally decreases faster as h–>0.

The methods are examples of a class of higher-order methods known as Runge–Kutta methods.

Derivation of the Midpoint Method:

Illustration of numerical integration for the equation y’ = y, y(0) =1. Blue: the Euler method, green: the midpoint method, red: the exact solution, y = exp(t) The step size is h = 1.0.

The same illustration for h = 0.25 It is seen that the midpoint method converges faster than the Euler method.

The midpoint method is a refinement of the Euler’s method

y_{n+1} = y_n + hf(t_n,y_n),\,

and is derived in a similar manner. The key to deriving Euler’s method is the approximate equality

y(t+h) \approx y(t) + hf(t,y(t)) \qquad\qquad (2)

which is obtained from the slope formula

<img class="thumbimage alignright" src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/24/Numerical_integration_illustration_step%3D0.25.svg/220px-Numerical_integration_illustration_step%3D0.25.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Numerical_integration_illustration_step%3D0.25.svg/330px-Numerical_integration_illustration_step%3D0.25.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Numerical_integration_illustration_step%3D0.25.svg/440px-Numerical_integration_illustration_step%3D0.25.svg.png 2x" alt="" width="220" height="320" data-file-width="440" data-file-height="640" /> <img class="mwe-math-fallback-image-inline aligncenter" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ffe42c2f261baa3d4b66630e185997f4f96579" alt=" y" aria-hidden="true" />

and keeping in mind that y’ = f(t,y)For the midpoint methods, one replaces (3) with the more accurate

y

when instead of (2) we find

y(t+h) \approx y(t) + hf\left(t+\frac{h}{2},y\left(t+\frac{h}{2}\right)\right). \qquad\qquad (4)

One cannot use this equation to find y(t+h) as one does not know y at t + h/2. The solution is then to use a Taylor series expansion exactly as if using the Euler method to solve for y(t+h/2):

y\left(t + \frac{h}{2}\right) \approx y(t) + \frac{h}{2}y

which, when plugged in (4), gives us

y(t + h) \approx y(t) + hf\left(t + \frac{h}{2}, y(t) + \frac{h}{2}f(t, y(t))\right)

and the explicit midpoint method (1e).

The implicit method (1i) is obtained by approximating the value at the half step t + h/2 by the midpoint of the line segment from y(t) to y(t+h)

y\left(t+\frac h2\right)\approx \frac12\bigl(y(t)+y(t+h)\bigr)

and thus

\frac{y(t+h)-y(t)}{h}\approx y

Inserting the approximation yn + h.k for y(tn+h) results in the implicit Runge-Kutta method

\begin{align} k&=f\left(t_n+\frac h2,y_n+\frac h2 k\right)\\ y_{n+1}&=y_n+h\,k \end{align}

which contains the implicit Euler method with step size h/2 as its first part.

Because of the time symmetry of the implicit method, all terms of even degree in h of the local error cancel, so that the local error is automatically of order O(h^3). Replacing the implicit with the explicit Euler method in the determination of k results again in the explicit midpoint method.

References:

[1] Griffiths,D. V.; Smith, I. M. (1991). Numerical methods for engineers: a programming approach. Boca Raton: CRC Press. p. 218. ISBN 0-8493-8610-1.

[2] Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0-521-00794-1

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