Responding to the person "Updating Name": I tried your version of the code, but I get a message: "The initial conditions did not evaluate to an array of numbers of depth 1 on the spatial grid. Initial conditions for partial differential equations should be specified as scalar functions of the spatial variables". To make your code work I had to remove WorkingPrecision -> 40, but then I get an accuracy of no more than 4-11 digits, depending on the choice of MaxPoints and MinPoints, while increasing AccuracyGoal or PrecisionGoal resulted in an enormous slowing down. My feeling is that NDSolve does not tolerate multiprecision, at least not in adaptive calculations.

Concerning the pseudospectral discretization: isn't it practical for periodic boundary conditions only?

Responding to Michael Rogers: I'll try your suggestions, but I am not happy about "ExplicitRungeKutta". Is there any possibility to employ implicit methods, say BDF? Or, other robust implicit schemes, like for example Rosenbrock methods? Some of these are very handy for nonlinear problems, as they do not require Newton iterations. I noticed in some examples on the internet that LSODA was used in NDSolve, and LSODA is based on BDF, if I remember. Explicit methods are likely to cause stability problems. Unfortunately, I cannot find anywhere a complete list of available integrators and other options for NDSolve. Leslaw

Used "FixedStep" + StartingStepSize for a fixed time step and whichever method for the spatial grid. Ignore the error/warning message about StartingStepSize. You've already complained about error messages, and I have no defense for this one. (It seems StartingStepSize may be used for both temporal and spatial discretization, and it's okay for one but not the other in the examples below.)

Example 1:

\[Nu] = 0.01;
mysol = First[
  NDSolve[{D[u[x, t], t] == \[Nu] D[u[x, t], x, x] - 
      u[x, t] D[u[x, t], x], u[x, 0] == E^-x^2, u[-5, t] == u[5, t]}
   , u, {x, -5, 5}, {t, 0, 4}
   , Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid", 
       "MinPoints" -> 257, "MaxPoints" -> 257,
        "DifferenceOrder" -> 4},
     Method -> {"FixedStep", Method -> "ExplicitRungeKutta"}}, 
   StartingStepSize -> 1/2^13]]

Example 2:

mygrid =  Join[-5. + 10  Range[0, 48]/80, 1. + Range[1, 4  70]/70];
\[Nu] = 0.01;
mysol = First[
  NDSolve[{D[u[x, t], t] == \[Nu] D[u[x, t], x, x] - 
      u[x, t] D[u[x, t], x], u[x, 0] == E^-x^2, u[-5, t] == u[5, t]}
   , u, {x, -5, 5}, {t, 0, 4}
   , Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid", 
       "Coordinates" -> {mygrid}},
     Method -> {"FixedStep", Method -> "ExplicitRungeKutta"}}, 
   StartingStepSize -> 1/2^13]]

The following gives the distinct step sizes:

u["Coordinates"] /. mysol // Map@Differences // Map@DeleteDuplicates

Examples adapted from the [Method of Lines tutorial.]

I tried some settings for WorkingPrecision, but then, if I remember, I get a series of error messages stating that something is not a number. The messages are totally incomprehensible, like most of error messages in MATHEMATICA, giving no clues to where and why the errors occurred. Forgive my irritation, but I really don't understand why the error handling is so bad in MATHEMATICA. Any compiler for programming languages is usually able to provide a precise location of errors, together with suggestions for eliminating the errors. Why this can't be done in MATHEMATICA? This should be particularly easy given the fact that compilation/execution is limited to a clear sequence of, or even to single commands. Leslaw

Well, thanks, this should be useful for output purpose, but my main concern and question is how to force NDSolve to produce results that have a precision of number representation higher than the standard 16 digits, and how to activate an effective use of higher order accurate discretizations (hopefully) built-into the procedure. My goal is to obtain the solution having many accurate digits, ideally 19-20 digits. This cannot be normally achieved by using standard floating point numbers and discretizations that are only 2nd or even 4th order accurate. Leslaw