OdeInt

A swift implementation of explict runge-kutta ODE integration using the Dormond Price RK5(4) method, also known as the DOPRI method. This uses adaptive time-stepping, and dense output to the times specified by the user.

The original method is documented in [1] as method 7M, and the dense output is documented in [2], Section 4.

Usage

The protocol OdeVector defines a state vector to be used in the integration, and both the standard SIMD types as well as the Double and Float types have been extended to conform to this protocol. Thus you can solve the ODE given by $dy/dx = -y$ with initial condition $y(0) = 1.0$ with the following:

let times = Array<Double>(stride(from: 0.0, through: 1.0, by: 0.001))
let results = Double.integrate(over: times, y0: 1.0, tol: 1e-6) { y, t in
  return -y
}

Or you can solve the same ODE using a vector initial condition $y(0) = [1, 2]$ using:

let times = Array<Float>(stride(from: 0.0, through: 1.0, by: 0.001))
let results = SIMD2<Float>.integrate(over: times, y0: [1.0, 2.0], tol: 1e-6) { y, t in
  return -y
}

References

[1] Dormand, J. R., & Prince, P. J. (1980). A family of embedded Runge-Kutta formulae. Journal of computational and applied mathematics, 6(1), 19-26.

[2] Dormand, J. R., & Prince, P. J. (1986). Runge-Kutta triples. Computers & Mathematics with Applications, 12(9), 1007-1017.