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equation solving - 4th-order Runge-Kutta method to solve a system of coupled ODEs




I am a beginner at Mathematica programming and with the Runge-Kutta method as well. I'm trying to solve a system of coupled ODEs using a 4th-order Runge-Kutta method for my project work.


I have solved it by NDSolve, but I want to solve this by 4th-order Runge-Kutta method. Here is my problem:


Γ = 1.4    
k = 0
z = 0
β = 0.166667

k1 = (d[η] v[η] η (1 - z d[η]) (v[η] - η) - 2 p[η] η (1 - z d[η]) - ϕ[η]^2 d[η] 
     (1 - z d[η]) - Γ p[η] v[η])/((Γ p[η] - (v[η] - η)^2 d[η] (1 - z d[η])) η)


k2 = (d[η] (1 - z d[η]) (v[η] d[η] (v[η] - 2 η) (v[η] - η) + 2 p[η] η + ϕ[η]^2 
     d[η]))/((Γ p[η] - (v[η] - η)^2 d[η] (1 - z d[η])) (v[η] - η) η)

k3 = (p[η] d[η] (2 η (v[η] - η)^2 (1 - z d[η]) + Γ v[η] (v[η] - 2 η) (v[η] - η) +
     ϕ[η]^2 Γ))/((Γ p[η] - (v[η] - η)^2 d[η] (1 - z d[η])) (v[η] - η) η)

k4 = -((ϕ[η] (v[η] + η))/(η (v[η] - η)))

k5 = -(w[η]/(η (v[η] - η)))


sol = NDSolve[{v'[η] == k1, d'[η] == k2, p'[η] == k3, ϕ'[η] == k4, w'[η] == k5, 
   v[1] == (1 - β), d[1] == 1/β, p [1] == (1 - β), ϕ[1] == 0.01, w[1] == 0.02}, 
  {v, d, p, ϕ, w}, {η, 0, 1}, MaxSteps -> 30000]

Please guide me how can I solve the above problem with 4th-order Runge-Kutta method, thanks.


code for RK4 method are given in Solving a system of ODEs with the Runge-Kutta method


but how can I apply those codes to my problem...please guide me...



Answer



According to your statement, I think what you need is just 4th-order Runge-Kutta method, and a completely self-made implementation of 4th-order Runge-Kutta method isn't necessary, then the answer from J.M. has shown you the optimal direction:


(* Unchanged part omitted. *)


ClassicalRungeKuttaCoefficients[4, prec_] :=With[{amat = {{1/2}, {0, 1/2}, {0, 0, 1}},
   bvec = {1/6, 1/3, 1/3, 1/6}, cvec = {1/2, 1/2, 1}}, N[{amat, bvec, cvec}, prec]]

sol = NDSolve[{v'[η] == k1, d'[η] == k2, p'[η] == k3, ϕ'[η] == k4, w'[η] == k5, 
   v[1] == (1 - β), d[1] == 1/β, p[1] == (1 - β), ϕ[1] == 0.01, w[1] == 0.02}, 
   {v, d, p, ϕ, w}, {η, 0, 1}, 
  Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4, 
    "Coefficients" -> ClassicalRungeKuttaCoefficients}, StartingStepSize -> 1/10000]


However, what I really want to point out is, despite the above code seems to solve your ODE set up to η = 0.0001, I'm afraid it's not reliable at all:


 {{nl, nr}} = (v /. sol)[[1]]["Domain"];
 Plot[{v@n, d@n, p@n, ϕ@n, w@n} /. sol // Evaluate, {n, nl, nr}]

enter image description here


NDSolve by default setting doesn't manage to solve this set of equation, too. It stopped at about η = 0.9576. (I'm not sure what do you mean by saying you have solved it by NDSolve.) I'm not surprised though, your ODEs are non-linear. As for how to solve the ODEs, it's another question. I vote to close this question as a duplicate.


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