Specifying mesh in NDSolve

Question

I am trying to solve a system of one-dimensional two-point boundary-value problems with NDSolve. I would like use a fixed mesh (specified by me) in the calculation. Is there a way to do this? The mesh can be uniformly spaced. Thanks!

Answer 1

First the mesh where you want NDSolve to solve the ODE-BVP.

(*Specifyling grid*)
{start, end, dist} = {0, 1, 1/10};
SuppliedMesh = N@Range[start, end, dist]

{0., 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.}

Now lets solve an example system below on the specified grid

(* Solving and saving the grid NDSolve uses *)
{sol, {grid}} = 
Reap[y /. Flatten@
 NDSolve[{y''[t] + y[t]/4 == 8, y[start] == 0, y[end] == 0}, 
  y, {t, start, end}, StartingStepSize ->dist, 
  EvaluationMonitor :> Sow[t],Method -> {"FixedStep", Method -> "ExplicitEuler"}]
];
(* Value of the interpolating function on its underlying mesh*)
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
InterpolatingFunctionValuesOnGrid[sol]

{0., -0.366755, -0.653511, -0.85935, -0.983554, -1.02561, -0.985208, -0.862242, -0.656812, -0.369227, 1.66533*10^-16}

Checking if the mesh matches really the supplied one.

SuppliedMesh === grid === Flatten@InterpolatingFunctionGrid[sol]

True

PS:

Now in place of supplying the mesh to NDSolve better one should try to make NDSolve make/use it in run-time while solving the equation by tuning its stepping algorithm. You will find such trick in NDSolvePlugIns.

BR