Explicit integration of PDE running very slowly

Question

I would like to solve the PDE $\frac{\partial u}{\partial t}=(1-x)\frac{\partial u}{\partial x}$, in the interval $[0,1]$, with initial condition $u(0,x)=e^{-218(\frac{-2}{3}-ln(1-x))^2}$, by manually implementing a pseudospectral method with Chebyshev nodes for spatial discretization and a Taylor method for time integration. The first ListPlot appears in a matter of seconds, but after that the code keeps running forever, to the point that I do not even know if it produces the correct results at the end. Is the code correct, and if so, can it become quicker? Please let me know for any clarifications.

a=218;
b=-2/3;
x0=0;
x1=1;
u[x_,t_,a1_,b1_]:=Exp[-a1*(b1+t-Log[1-x])^2];
f[x_]:=u[x,0,a,b];
D[t,u]-(1-x)*D[x,u]
der1[x0_]:=NDSolveFiniteDifferenceDerivative[Derivative[1],x0, "DifferenceOrder"->"Pseudospectral"]@"DifferentiationMatrix"; Nx=2^8; II=IdentityMatrix[Nx]; n=Nx-1; theta=N[Table[i*Pi/n,{i,0,n}]]; X=N[Chop[((x0+x1)/2)+((x0-x1)/2)*Cos[theta]]]; dt=N[0.01*(x1-x0)/n]; D1=der1[X]; L=(1-x)*D1; L2=L.L; L3=L2.L; M = (II+dt*L+((dt^2)/2)*L2+((dt^3)/6)*L3); M= DeveloperToPackedArray[M, Real];
Nt=10^5;
T=Table[it*dt,{it,0,Nt-1}];
U=ConstantArray[0.,{Nt, Nx}];
U[[1, All]]=f[X];
U= Developer`ToPackedArray[U, Real];
ListPlot[Transpose[{X, U[[1, All]]}], PlotRange->All, Joined->True]
Do[U[[it+1,All]]=M.U [[it,All]],{it,1,Nt-1}]
ListPlot[Transpose[{X,U[[Nt, All]]}], PlotRange->All, Joined->True]
ListPlot[Transpose[{X,U[[Nt, All]]-f[X-T[[Nt]]]}], PlotRange->All, Joined->True]

Answer 1

There are typos in the code that x in L = (1 - x)*D1 is not defined and also in the initial data last point is Indeterminate. General structure of the code should be revisited as follows

Clear["Global`*"]
Needs["Developer`"]
b = -2/3; a = 218;
x0 = 0;
x1 = 1;
u[x_, t_, a1_, b1_] := Exp[-a1(b1 + t - Log[1 - x])^2];
f[x_] := u[x, 0, a, b];
D[u, t] - (1 - x)D[u, x]
der1[x0_] := 
  NDSolveFiniteDifferenceDerivative[Derivative[1], x0, "DifferenceOrder" -> "Pseudospectral"]@"DifferentiationMatrix"; Nx = 2^8; II = IdentityMatrix[Nx]; n = Nx - 1; theta = N[Table[i*Pi/n, {i, 0, n}]]; X = N[Chop[((x0 + x1)/2) + ((x0 - x1)/2)*Cos[theta]]]; dt = N[0.01*(x1 - x0)/n]; D1 = der1[X]; L = (II - DiagonalMatrix[X]) . D1; L2 = L . L; L3 = L2 . L; M = (II + dt*L + ((dt^2)/2)*L2 + ((dt^3)/6)*L3); M = DeveloperToPackedArray[M, Real];
Nt = Round[1/dt];
T = Table[it*dt, {it, 1, Nt}];
U[0] = Chop[f[X] /. Indeterminate -> 0] // Quiet;

Iteration on time takes about 0.25 s

Do[U[it] = M . U[it - 1], {it, 1, Nt}]

Visualization of function and logarithm of absolute error on several steps

    Table[{ListPlot[Transpose[{X, U[i]}], PlotRange -> All, 
   Joined -> True, PlotLabel -> i dt],
  ListPlot[Transpose[{X, Abs[U[i] - u[X, T[[i]], a, b] // Quiet]}], 
   PlotRange -> All, Joined -> True, ScalingFunctions -> "Log"]}, {i, 
  1, Nt, Round[Nt/10]}]