Numerical Approximation of a 4D System of ODE's

[TylerH]

What is the most general method of approximating arbitrary systems of ODEs of 4 variables(x,y,z,t) that fit these conditions? The conditions that are assumed true of the ODEs are:
1) that I require differentials to be explicitly defined (but they can be defined in terms of other differentials).
2) that time is assumed linear (ei, it's a 3 variable system described in terms of 4, but the 4th is known)

That is:
dx=y^2+sin(x)
dy=-dz+y*t
dz=t*z

would be valid, because dx,dy, and dz are explicitly defined and alone on one side.

dx=f(dx)

would be invalid, because I don't want to solve implicit equations.

f(dx,x,y,z,t) = g(x,y,z,t)

would be invalid, because I don't want to have to move all the junk to the other side to get an explicit definition of dx.

 

[bigfooted]

A (4-stage) Runge-Kutta method usually works pretty well for systems of explicit coupled ode's that are not too stiff. If your equation becomes very stiff, you probably need to switch to specialized stiff solvers, which are usually (semi-)implicit.

 

[TylerH]

What are some implicit methods that you would recommend? With generality being the most favored quality. I'm okay with the algorithm being harder and slower, it's all going to be precomputed anyway.

When I said I didn't want to solve implicit equations, I was only really talking about avoiding symbolic math.

Thanks for the response.

 

[bigfooted]

You could start with the backward Euler method, it is unconditionally stable and the easiest of a large class of implicit methods (adams-moulton)

 

[blue_raver22]

The most general method for approximating arbitrary systems of ODEs of 4 variables (x,y,z,t) that fit these conditions would be to use numerical approximation techniques. These techniques involve breaking down the system of ODEs into smaller, simpler equations and using numerical methods such as Euler's method, Runge-Kutta method, or the Adams-Bashforth method to approximate the solutions.

In this case, the system of ODEs can be rewritten as:

dx/dt = f(x,y,z,t)
dy/dt = g(x,y,z,t)
dz/dt = h(x,y,z,t)

where f, g, and h are functions that define the differentials in terms of the other variables. With this form, we can use any of the numerical methods mentioned above to approximate the solutions for x, y, z, and t at different time intervals.

It's important to note that the accuracy of the numerical approximation will depend on the step size chosen and the complexity of the equations. A smaller step size will result in a more accurate approximation, but it will also require more computations. On the other hand, a larger step size will be less accurate but will require fewer computations.

In conclusion, the most general method for approximating a 4D system of ODEs would be to use numerical approximation techniques, such as Euler's method, Runge-Kutta method, or the Adams-Bashforth method. These methods allow for the explicit definition of differentials and can handle linear time assumptions. The accuracy of the approximation will depend on the chosen step size and the complexity of the equations.