Finite Volume Solution Reconstruction Methods For Truncation Error Estimation

Abstract

The numerical solution to differential equations results in a discrete solution space for the finite volume and finite difference discretization methods. For various reasons, it can be necessary to prolong the solution from a discrete space to a continuous space. The prolongation to a continuous space can be done using various curve-fitting methods which adds an additional level of approximation to the solution. The allowable error of a prolongation operation depends on the specific task required by the user. In this paper we investigate various prolongation methods and identify the minimal requirements specifically for the purpose of truncation error estimation (the difference between the discrete and integral governing equations) for finite volume methods. The reconstruction methods investigated will include k-exact and ENO methods. Truncation error estimation for 1D Burgers’ equation and the k-exact method suggest that the minimum polynomial order is dependent on the highest derivatives in the truncation error expression and, therefore, the discretization scheme. The effect of different reconstruction methods on truncation error estimation is investigated and the minimum polynomial order for accurate truncation error estimation is identified for the Euler equations and is found to be second-order for the weak formulation and third-order for the strong formulation.