A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes (Advances in Applied Mathematics and Mechanics, 2018)
We modify the construction of the third order finite volume WENO schemeon triangular meshes and present a simplified WENO (SWENO) scheme. The noveltyof the SWENO scheme is the less complexity and lower computational cost whendeciding the smoothest stencil through a simple mechanism. The LU decompositionwith iterative refinement is adopted to implement ill-conditioned interpolation matricesand improves the stability of the SWENO scheme efficiently. Besides, a scaling techniqueis used to circumvent the growth of condition numbers as mesh refined. However,weak oscillations still appear when the SWENO scheme deals with complex lowdensity equations. In order to guarantee the maximum-principle-preserving (MPP)property, we apply a scaling limiter to the reconstruction polynomial without the lossof accuracy. A novel procedure is designed to prove this property theoretically. Finally,numerical examples for one- and two-dimensional problems are presented to verifythe good performance, maximum principle preserving, essentially non-oscillation andhigh resolution of the proposed scheme.