A numerical boundary integral method, relating slip and traction on a plane in an elastic medium by convolution with a discretized Green function, can be linked to a slip-dependent friction law on the fault plane. Such a method is developed here in two-dimensional plane-strain geometry. The method is more efficient for a planar source than a finite difference method, and it does not suffer from dispersion of short wavelength components. The solution for a crack growing at constant velocity agrees closely with the analytic solution, and the energy absorbed at the smeared-out crack tip in the numerical calculation agrees with energy absorbed at the analytic singularity. Spontaneous plane-strain shear ruptures can make a transition from sub-Rayleigh to near-P propagation velocity. Results from the boundary integral method agree with earlier results from a finite difference method on the location of this transition in parameter space. The methods differ in their prediction of rupture velocity following the transition. The trailing edge of the cohesive zone propagates at the P-wave velocity after the transition in the boundary integral calculations.