The Application of Iterated Defect Corrections Based on WENO Reconstruction

Alexander Filimon and Claus-Dieter Munz

Abstract. In this article we apply the procedure of the iterated defect correc­tion method to the Euler equations as well as to the Navier-Stokes equations. One building block in the defect correction approach is the lower order basic method, usually first or second order accurate. This scheme gives a steady solution of low accuracy as the starting point. The second building block is the WENO reconstruc­tion step to estimate the local defect. The local defect is put into the original equation as source on the right hand side with a minus sign. The resulting modified equation is then again solved with the low order scheme. Due to the source term with the local defect the order of accuracy is iteratively shifted to the order of the reconstruction. We show numerical results for several validation test cases and applications.

1 Introduction

Numerical simulations of the equations of fluid mechanics contain unavoidable er­rors due to several necessary approximations. To analyze these errors is crucial for the evaluation of the reliability of the numerical results. In the following we fo­cus ourselves to the discretization errors. This means, that the modeling errors are excluded and the exact solution of the governing equations is supposed to be the reference solution of the described physical phenomenon.

The discretization errors can be separated into local and global discretization errors. By inserting the exact solution into the discretized equations, the local dis­cretization error, also known as the local defect of a numerical approximation, can be determined. The more significant global discretization error gives the difference

Alexander Filimon • Claus-Dieter Munz

Institute of Aerodynamics and Gas Dynamics, 70569 Stuttgart, Germany e-mail: {filimon, munz}@iag. uni-stuttgart. de

B. Eisfeld et al. (Eds.): Management & Minimisation of Uncert. & Errors, NNFM 122, pp. 129-154. DOI: 10.1007/978-3-642-36185-2_6 © Springer-Verlag Berlin Heidelberg 2013

between the numerical and the exact solution. In both cases the exact solution is needed, which makes the error approximation for real applications cumbersome. A common approach is to run the same problem on several meshes with differ­ent step size h. A mesh convergence study allow then to compute the so called experimental convergence rate. Finally, a Richardson extrapolation can be used to determine the best approximate solution together with an estimation of the global discretization error. In practical 3D applications with complex geometries, this ap­proach becomes cumbersome and sometimes even unfeasible because of the high computational costs. Our approach allows an error approximation for steady prob­lems on the original mesh by using a polynomial reconstruction within the defect correction method.

Starting with a steady solution of a first or second order accurate finite volume scheme, we employ the modified weighted essentially non oscillatory (WENO) re­construction scheme of Dumbser and Kaser [4] for unstructured meshes. The res­ulting polynomial distribution allows an improved flux computation which can be applied to estimate the local discretization error. The method of the iterated defect correction (IDeC) consists of subtracting this local defect as a source term on the right hand side of the original equations [23, 19]. The now modified equations are solved with the original method of first or second order accuracy, in the following also called the basic method or the basic scheme, resulting into a new corrected steady solution. A further reconstruction of the corrected solution yields a better estimation of the local defect which is now used in the modified equations. Iterat­ively applied, the method of the defect correction shifts the order of accuracy of the basic scheme to the higher order of the used reconstruction. By this approach, an approximation of the global discretization error up to an accuracy of the higher order reconstruction is available.