Development of a Stable High-Order Point-Value Enhanced Finite Volume (PFV) Method Based on Approximate Delta Functions

摘要

In this work, we generalize the expression of an approximate delta function (ADF), which is a finite-order polynomial that holds identical integral properties to the Dirac delta function, particularly, when used in conjunction with a finite-order polynomial integrand over a finite domain. By focusing on one-dimensional configurations, we show that the use of generalized ADF polynomials can be effective at recovering and extending several high-order methods, including Taylor-based and nodal-based Discontinuous Galerkin methods, as well as the Correction Procedure via Reconstruction. Based on the ADF concept, we then proceed to formulate a Point-value enhanced Finite Volume (PFV) method, which stores and updates the cell-averaged values inside each element as well as the unknown quantities and, if needed, their derivatives on nodal points. The sharing of nodal information with surrounding elements reduces the number of degrees of freedom compared to other compact methods at the same order. To ensure conservation, cell-averaged values are updated using an identical approach to that adopted in the finite volume method. Presently, the updating of nodal values and their derivatives is achieved through an ADF concept that leverages all of the elements within the domain of integration that share the same nodal point. The resulting scheme is shown to be very stable at successively increasing orders. Both accuracy and stability of the PFV method are verified using a Fourier analysis and through applications to two benchmark cases, namely, the linear wave and nonlinear Burgers' equations in one-dimensional space.