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Hello!
An assignment for my computational modeling course is to demonstrate the use of the Standard Euler method for modeling a simple harmonic oscillator; in this case, a mass attached to the end of a spring.
I have the two coupled firstorder differential equations satisfying hookes law: dx/dt = v, and dv/dt = (k/m)*x
The numerical solutions of which are v(t + dt) = v(t)  k*x(t)*dt / m, and x(t + dt) = x(t) + v(t)*dt to model the velocity and position, respectively.
In a computer program, I've represented this as:
x = x + v*dt;
v = v  k*x*dt / m;
t = t + dt;
with dt = 0.04, m = 1, k = 1, and initial values v = 0, t = 0, and x = 5.
The Purpose of the exercise is to demonstrate how the standard Euler Method is nonstable and results in nonconserved energy.
When iterating the above Euler method for sufficiently large periods of time, I've expected x to grow larger after each period but my numerical method above is acting like a conservedenergy Improved Euler method (Eulercromer)? Please see attached plot.
An assignment for my computational modeling course is to demonstrate the use of the Standard Euler method for modeling a simple harmonic oscillator; in this case, a mass attached to the end of a spring.
I have the two coupled firstorder differential equations satisfying hookes law: dx/dt = v, and dv/dt = (k/m)*x
The numerical solutions of which are v(t + dt) = v(t)  k*x(t)*dt / m, and x(t + dt) = x(t) + v(t)*dt to model the velocity and position, respectively.
In a computer program, I've represented this as:
x = x + v*dt;
v = v  k*x*dt / m;
t = t + dt;
with dt = 0.04, m = 1, k = 1, and initial values v = 0, t = 0, and x = 5.
The Purpose of the exercise is to demonstrate how the standard Euler Method is nonstable and results in nonconserved energy.
When iterating the above Euler method for sufficiently large periods of time, I've expected x to grow larger after each period but my numerical method above is acting like a conservedenergy Improved Euler method (Eulercromer)? Please see attached plot.
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