Enforcing causality in numerical solutions of Maxwell's equations

Maxwell's equations in differential form do not distinguish between advanced and retarded solutions. Unless special precautions are taken, a point-by-point numerical integration, based on a finite-difference analog of Maxwell's equations, will lead to a mixture of advanced and retarded fields, inadmissible on physical grounds. The causality requirement can be satisfied if the Maxwell theory is expressed in integral-equation form, with retardation incorporated in all the integrands. A solution using numerical integration will then be physically acceptable.