Numerical solution of a parabolic equation subject to specification of energy
In this paper special attention is devoted to one particular problem, the diffusion equation subject to the specification of mass in a portion of the domain, which has been studied quite extensively, both analytically and numerically, in recent years. Parabolic partial differential equations with a non-local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena. In this paper the application of the method of lines (MOL) to such problems is considered. The MOL semi-discretization approach will be used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs). The MOL is a method of solving PDEs by discretizing the equation with respect to all but one variable (usually time). The spatial partial derivative is approximated by a finite-difference method. The solution of the resulting system of first-order ODEs satisfies a recurrence relation which involves a matrix exponential function. Numerical techniques are developed by approximating the exponential matrix function in this recurrence relation. The matrix exponential function is approximated by a rational approximation consisting of four parameters. New parallel algorithms are developed using the resulting approximation. Numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms.