  We present fast, spatially dispersionless and unconditionally stable
high-order solvers for Partial Differential Equations (PDEs) with variable
coefficients in general smooth domains. Our solvers, which are based on (i) A
certain "Fourier continuation" (FC) method for the resolution of the Gibbs
phenomenon, together with (ii) A new, preconditioned, FC-based solver for
two-point boundary value problems (BVP) for variable-coefficient Ordinary
Differential Equations, and (iii) An Alternating Direction strategy, generalize
significantly a class of FC-based solvers introduced recently for
constant-coefficient PDEs. The present algorithms, which are applicable, with
high-order accuracy, to variable-coefficient elliptic, parabolic and hyperbolic
PDEs in general domains with smooth boundaries, are unconditionally stable, do
not suffer from spatial numerical dispersion, and they run at FFT speeds. The
accuracy, efficiency and overall capabilities of our methods are demonstrated
by means of applications to challenging problems of diffusion and wave
propagation in heterogeneous media.
