  We present an improved spectral algorithm for Cauchy-characteristic
extraction and characteristic evolution of gravitational waves in numerical
relativity. The new algorithms improve spectral convergence both at the poles
of the spherical-polar grid and at future null infinity, as well as increase
the temporal resolution of the code. The key to the success of these algorithms
is a new set of high-accuracy tests, which we present here. We demonstrate the
accuracy of the code and compare with the existing PittNull implementation.
