Efficient Filter-Enhanced Polynomial Collocation Method for Solving Discontinuous and Steep Gradient Problems

The presence of irregularities and steep gradients in functions is known to degrade the accuracy and stability of numerical approximations. To overcome this, a filtering-based methodology is proposed in this study with minimal added complexity. An ultraspherical polynomial spectral collocation framework is employed, and filtered differentiation matrices are formulated at Gauss-Lobatto points. The centro-antisymmetric structure of these matrices is exploited, by which the computational effort is nearly halved. Two representative problems are examined: a discontinuous function and a variable-coefficient wave equation. The unfiltered approximation of the latter is found to be unstable at larger time levels. For the wave equation, the resulting system of ODEs is integrated using the classical fourth-order Runge-Kutta method, and a stability analysis is presented. The results show that stability and accuracy are markedly improved by spectral filtering. Faster and more effective convergence is achieved through Chebyshev filtering than through Legendre filtering.