Energy-Conserving Hermite Methods for Maxwell’s Equations

Abstract Energy-conserving Hermite methods for solving Maxwell’s equations in dielectric and dispersive media are described and analyzed. In three space dimensions, methods of order 2 m to $$2m+2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> require $$(m+1)^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of m . We prove the stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special semi-norm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of the electromagnetic wave propagation over thousands of wavelengths.