High accuracy time and space transform method for advection-diffusion equation in an unbounded domain

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection– diffusion equation in the unbounded domain Rn is converted to one on the bounded domain [−1,1]n , and the Laplace transform is applied to eliminate time dependency. The consequent boundary value problem is solved by collocation on Chebyshev points. To face the well-known computational challenge represented by the numerical inversion of the Laplace transform, Talbot’s method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that TSTM has exponential rate in time and space. Copyright q 2008 John Wiley & Sons, Ltd.