A Framework for Arbitrary-Order Electromagnetic Field Approximation Based on Analytical Solutions of Maxwell's Equations

The standard Yee finite-difference time-domain (FDTD) algorithm uses Maxwell’s equations to replace first order temporal derivatives with spatial curls. In this work, the relation is extended to arbitrary orders: a Time-Space Theorem establishes the correspondence between higher-order temporal derivatives and successively higher-order spatial curl operators. Based on this relation, an arbitrary-order framework for electromagnetic field approximations is developed. The framework constructs higher temporal orders through higher curl orders and permits different spatial discretizations and curl-estimation techniques, thereby providing a general basis for developing numerical algorithms of arbitrary orders. As one implementation, an arbitrary-order FDTD algorithm is constructed using staggered time and space stepping together with recursive curl estimations. Numerical simulations of an initial-value problem are performed for temporal approximation orders from second to tenth order while the spatial and temporal step sizes are kept fixed. Comparisons with the analytical solution show that the errors decrease rapidly and monotonically with increasing approximation order. These results demonstrate the practical potential of the framework for high-accuracy electromagnetic computation and provide a basis for further development of arbitrary-order numerical methods.