Generalized alternating‐direction collocation methods for parabolic equations. I. Spatially varying coefficients

The alternating‐direction collocation (ADC) method can be formulated for general parabolic partial differential equations. This is done using a piecewise cubic Hermite trial space defined on a rectangular discretization. As in all alternating‐direction methods, the ADC algorithm produces errors that are additional to the standard discretization errors of multi‐dimensional collocation. These errors increase when the coefficients of the governing equation are spatially variable. Analysis of the additional errors leads to several correction schemes. Numerical results indicate that a variant on the Laplace‐modification procedure is an attractive choice as an improved ADC algorithm.