Vrftjkry

I tried to NDSolve the PDE system
$$partial_t w =xcdot wquadquadpartial_z x=w$$
for
$$(t,z)in[0,1]times[0,pi]$$
with boundary conditions
$$x(t,0)=w(t,0)=w(t,pi)=0$$
and initial conditions
$$w(0,z)=sin zquadquad x(0,z)=1-cos z$$
Here's my code:

s = NDSolve[{D[w[t, z], t] == w[t, z]*x[t, z], 

   D[x[t, z], z] == w[t, z], w[0, z] == Sin[z], x[0, z] == 1 - Cos[z],

    w[t, 0] == 0, w[t, π] == 0, x[t, 0] == 0}, {w , x}, {t, 0, 

   1}, {z, 0, π}]

Mathematica displays the following warning:

"NDSolve::mconly: For the method NDSolve`IDA, only machine real code
is available. Unable to continue with complex values or beyond
floating-point exceptions."

I would appreciate any help on how to overcome this error or solve numerically this kind of PDE system anyway.

This is a quasilinear hyperbolic system of equations. Not all initial data is valid, w=0 should be excluded from the initial data. An example of solving the problem

s = NDSolve[{D[w[t, z], t] == w[t, z]*x[t, z], 

   D[x[t, z], z] == w[t, z], w[0, z] == 1, x[0, z] == 1 - Cos[z], 

   w[t, 0] == 1, w[t, [Pi]] == 1, x[t, 0] == 0}, {w, x}, {t, 0, 

   1}, {z, 0, [Pi]}, 

  Method -> {"MethodOfLines", 

    "SpatialDiscretization" -> {"TensorProductGrid", 

      "MinPoints" -> 80, "MaxPoints" -> 100, 

      "DifferenceOrder" -> "Pseudospectral"}}];

{ContourPlot[Evaluate[w[t, z] /. s], {t, 0, 1}, {z, 0, [Pi]}, 

  Contours -> 20, ColorFunction -> Hue, PlotLabel -> "w", 

  FrameLabel -> {"t", "z"}, PlotLegends -> Automatic], 

 ContourPlot[Evaluate[x[t, z] /. s], {t, 0, 1}, {z, 0, [Pi]}, 

  Contours -> 20, ColorFunction -> Hue, PlotLabel -> "x", 

  FrameLabel -> {"t", "z"}, PlotLegends -> Automatic]}