  We present an overarching framework for stable spectral methods on a
triangle, defined by a multivariate W-system and based on orthogonal
polynomials on the triangle. Motivated by the Koornwinder orthogonal
polynomials on the triangle, we introduce a Koornwinder W-system. Once
discretised by this W-system, the resulting spatial differentiation matrix is
skew symmetric, affording important advantages insofar as stability and
conservation of structure are concerned. We analyse the construction of the
differentiation matrix and matrix vector multiplication, demonstrating optimal
computational cost. Numerical convergence is illustrated through experiments
with different parameter choices. As a result, our method exhibits key
characteristics of a practical spectral method, inclusive of rapid convergence,
fast computation and the preservation of structure of the underlying partial
differential equation.
