The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources

We extend the immersed interface method of LeVeque and Li to find numerical solutions of one-dimensional parabolic partial differential equations of the form u(t) = (beta(x, t)u(x))(x) + (lambda(x, t)u)(x) + kuu(x) f(x), where beta, u, beta ux, and f may have known discontinuities at a known location x = alpha. At each time step, a large, weakly nonlinear system is set up using a difference scheme which is standard away from x = alpha and which is derived for grid points near alpha by solving small linear systems which are determined from the jumps at x = alpha. The time-stepping is done with a Crank-Nicholson scheme, and the nonlinear systems are solved with a Levenberg-Marquardt method. As an example, we consider the ow of cars on a one-lane highway with an entrance or exit, where traffic is treated as a continuous fluid. Numerical examples show that we can compute solutions to these equations with second-order accuracy.