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 uₜ = (β(x,t) uₓ)ₓ + (λ(x,t) u)ₓ + κu uₓ-f(x), where β, u, β uₓ, and f may have known discontinuities at a known location x = α. At each time step, a large, weakly nonlinear system is set up using a difference scheme which is standard away from x = α and which is derived for grid points near α by solving small linear systems which are determined from the jumps at x = α. 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 flow 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.