Spectral methods on a triangle and W-systems

Abstract

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.