Maxwell's electromagnetic equations coupled to Fourier's heat transfer equation in a three-dimensional space are numerically solved together to simulate the microwave heating of a dielectric material inside a multi-mode cavity. The numerical scheme is based on the method of lines which is a variant of the traditional finite difference time domain technique. A detailed derivation is given of the system of ordinary differential equations which describe the electric and magnetic fields. A new procedure is established to impose conditions on a power entry port. Changes to the standard finite difference expressions are made to allow for the presence of dielectric interfaces, and the convergence criteria are derived and tested. The theoretical treatment is supported by examples of practical importance to industry.
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