# Solving equations using Implicit-Runge-Kutta method [closed]

I am trying to solve the system of equations using Implicit-Runge-Kutta method but mathematica is showing error "NDSolve::ndcf: Repeated convergence test failure at [Tau] == 4.9; unable to continue.". Can anyone help me please to solve this system?

Equations = {
Derivative[Xt][τ] == Ut[τ],
Xr'[τ] == Ur[τ],
Xθ'[τ] == Uθ[τ],
Xϕ'[τ] == Uϕ[τ],
(2 Ur[τ] Ut[τ])/((-2 + Xr[τ]) Xr[τ]) +
Derivative[Ut][τ] ==
0 , -Uθ[τ]^2 (-2 + Xr[τ]) -
Sin[Xθ[τ]]^2 Uϕ[τ]^2 (-2 + Xr[τ]) + (
Ut[τ]^2 (-2 + Xr[τ]))/Xr[τ]^3 + Ur[τ]^2/(
2 Xr[τ] - Xr[τ]^2) + Derivative[Ur][τ] ==
0 , -Cos[Xθ[τ]] Sin[
Xθ[τ]] Uϕ[τ]^2 + (
2 Ur[τ] Uθ[τ])/Xr[τ] +
Derivative[Uθ][τ] == 0 ,
2 Cot[Xθ[τ]] Uθ[τ] Uϕ[τ] + (
2 Ur[τ] Uϕ[τ])/Xr[τ] +
Derivative[Uϕ][τ] == 0 };
IntCond = {
Xt == 0, Xr == 0.3, Xθ == Pi/2 + 0.4,
Xϕ == 0,
Ut == 0.01, Ur == 0.001, Uθ == 0.001,
Uϕ == 0.1};
sol = NDSolve[{Equations, IntCond}, {Xt, Xr, Xθ, Xϕ, Ut,
Ur, Uθ, Uϕ}, {τ, 0, 1000},
Method -> {"FixedStep",
Method -> {"ImplicitRungeKutta", "DifferenceOrder" -> 4,
"ImplicitSolver" -> {"Newton", AccuracyGoal -> MachinePrecision,
PrecisionGoal -> MachinePrecision,
"IterationSafetyFactor" -> 1}}}, StartingStepSize -> 1/10]

• "some error": it would be helpful to include the type of error and details. – MarcoB Feb 12 at 15:47
• mathematica showing the error NDSolve::ndcf: Repeated convergence test failure at [Tau] == 4.9; unable to continue. – MMS Feb 12 at 15:49
• U\[Theta], U\[Phi] both seem to head to infinity around 4.953775. Try it with Method -> Automatic and plot each component of the solution. I think the solution has a singularity. – Michael E2 Feb 12 at 20:32
• I want to apply implicitRungeKutta method because using this method energy (constant of motion) will remain conserve throughout the motion, error will be negligible and results will be more accurate. But don't know how to remove error. – MMS Feb 12 at 20:50
• Even with Method -> "ImplicitRungeKutta", the error cannot be removed because the solution has a singularity around 4.95. It is not an error in the sense of a mistake that can be fixed or removed. It is a feature of the problem that cannot be avoided. If there should not be such a singularity, then perhaps the problem has been miscoded? – Michael E2 Feb 13 at 3:56