The Immersed Interface Method for Nonlinear Differential Equations with Discontinuous Coefficients and Singular Sources

• Authors: Andreas Wiegmann;Kenneth P. Bube
• Venue: SIAM Journal on Numerical Analysis
• Year: 1998

We extend the immersed interface method of LeVeque and Li to find numerical solutions of one-dimensional parabolic partial differential equations of the form $u_t = (\beta(x,t) u_x)_x + (\lambda(x,t) u)_x + {\kappa}u u_x-f(x)$, where $\beta$, u, $\beta u_x$, 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 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.