  We study coherent synchrotron radiation (CSR) in a perfectly conducting
vacuum chamber of rectangular cross section, in a formalism allowing an
arbitrary sequence of bends and straight sections. We apply the paraxial method
in the frequency domain, with a Fourier development in the vertical coordinate
but with no other mode expansions. A line charge source is handled numerically
by a new method that rids the equations of singularities through a change of
dependent variable. The resulting algorithm is fast compared to earlier
methods, works for short bunches with complicated structure, and yields all six
field components at any space-time point. As an example we compute the
tangential magnetic field at the walls. From that one can make a perturbative
treatment of the Poynting flux to estimate the energy deposited in resistive
walls. The calculation was motivated by a design issue for LCLS-II, the
question of how much wall heating from CSR occurs in the last bend of a bunch
compressor and the following straight section. Working with a realistic
longitudinal bunch form of r.m.s. length $10.4~\mu$m and a charge of 100 pC we
conclude that the radiated power is quite small (28 W at a 1 MHz repetition
rate), and all radiated energy is absorbed in the walls within 7 m along the
straight section.
