Solving multivariable mathematical models by the quadrature and cubature methods

The utilization and generalization of quadrature and cubature approximations for numerical solution of mathematical models of multivariable transport processes involving integral, differential, and integro-differential operators, and for numerical interpolation and extrapolation, are presented. The methodology for determination of the quadrature and cubature weights for composite operators is developed to accommodate for general functional representations. Application of these methods is demonstrated by solving two-dimensional steady-state and one-dimensional transient-state problems. The solutions are compared with exact-analytical solutions to evaluate the performance of these methods. It is demonstrated that the quadrature and cubature approximations are simple and universal; i.e., the same formula is applicable irrespective of the order of accuracy of the numerical approximation, the type of linear operator, and the number of temporal and/or spatial variables. Since the quadrature and cubature methods can produce solutions with sufficient accuracy even when using fewer discrete points, both the programming task and computational effort are reduced considerably. Therefore, the quadrature and cubature methods appear to be very practical in solving the mathematical models of a variety of transport processes. © 1994 John Wiley & Sons, Inc.