Rapid evaluation of two-dimensional retarded time integrals

We present two methods for rapid evaluation of two-dimensional retarded time integrals. For example, such integrals arise as the z=0 trace U(t,x,y,0) of a solution U(t,x,y,z) to 3+1 wave equation □U=−2f(t,x,y)δ(z) forced by a “sheet source” at z=0. The spatial Fourier transform of a two-dimensional retarded time integral involves a temporal convolution with the zeroth order Bessel function J0(t). Appealing to work by Alpert, Greengard, and Hagstrom and by Xu and Jiang on rational approximation in the Laplace-transform domain, our first method relies on approximation of J0(t) as a sum of exponential functions. We achieve approximations with double precision accuracy near t≃0, and maintain single precision accuracy out to T≃108. Our second method involves evolution of the 3+1 wave equation in a “thin block” above the sheet, adopting the radiation boundary conditions of Hagstrom, Warburton, and Givoli based on complete plane wave expansions. We review their technique, present its implementation for our problem, and present new results on the nonlocal spacetime form of radiation boundary conditions. Our methods are relevant for the sheet-bunch formulation of the Vlasov-Maxwell system, although here we only test methods on a model problem, a Gaussian source following an elliptical orbit. Our concluding section discusses the complexity of both methods in comparison with naive evaluation of a retarded-time integral.