Structure-preserving spectral methods on a triangle

Abstract We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials. Motivated by the Koornwinder orthogonal polynomials on the triangle we introduce a Koornwinder W-system. Once discretized by this W-system the resulting spatial partial differentiation matrices are skew-symmetric, affording important advantages insofar as stability and conservation of structure are concerned. We analyse the construction of the differentiation matrix. A major advantage of our approach is that it leads to linear algebraic systems with semiseparable matrices, which can be solved rapidly. Numerical performance is illustrated through experiments with different parameter choices. Our method exhibits key characteristics of a practical spectral method, exhibiting rapid convergence, fast linear algebra, stability and the preservation of structure of the underlying partial differential equation.