Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.