A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes

Computer Methods in Applied Mechanics and Engineering

et al. 2015

This paper is concerned with the extension of second order residual distribution (RD) schemes to time dependent viscous flows. We provide a critical analysis of the use of a hybrid RD-Galerkin approach for both steady and time dependent problems. In particular, as in Ricchiuto (2008) and Villedieu etal. (2011), we study the coupling of a Residual Distribution (RD) discretization of the advection operator with a Galerkin approximation for the second order derivatives, with a Peclet dependent modulation of the upwinding introduced by the RD scheme. The final objective is to be able to achieve uniform second order of accuracy with respect to variations of the mesh size or, equivalently, of the local Peclet/Reynolds number. Starting from the scalar formulation given in the second order case in Ricchiuto etal. (2008), we perform an accuracy and stability analysis to extend the approach to time-dependent problems, and provide thorough numerical validation of the theoretical results. The schemes, formally extended to the system of laminar Navier-Stokes equations, are also compared to a finite volume scheme with least-squares linear reconstruction on the solution of standard test problems. While the scalar tests show the potential of this approach in providing uniform accuracy w.r.t. the range of values of the Peclet/Reynolds number, the results for the laminar Navier-Stokes equations show that in practice, the use of Peclet number based corrections does not change dramatically the quality of the solutions obtained. This justifies the quest for new formulations. This paper is concerned with the extension of second order residual distribution (RD) schemes to time dependent viscous flows. We provide a critical analysis of the use of a hybrid RD-Galerkin approach for both steady and time dependent problems. In particular, as in (Ricchiuto et al. J.Comp.Appl.Math. 215, 2008, Villedieu et al. J.Comput.Phys, 230, 2011, we study the coupling of a Residual Distribution (RD) discretization of the advection operator with a Galerkin approximation for the second order derivatives, with a Peclet dependent modulation of the upwinding introduced by the RD scheme. The final objective is to be able to achieve uniform second order of accuracy with respect to variations of the mesh size or, equivalently, of the local Peclet/Reynolds number. Starting from the scalar formulation given in the second order case in (Ricchiuto et al. J.Comp.Appl.Math. 215, 2008), we perform an accuracy and stability analysis to extend the approach to time-dependent problems, and provide thorough numerical validation of the theoretical results. The schemes, formally extended to the system of laminar NavierStokes equations, are also compared to a finite volume scheme with least-squares linear reconstruction on the solution of standard test problems. While the scalar tests show the potential of this approach in providing uniform accuracy w.r.t. the range of values of the Peclet/Reynolds number, the results for the laminar Navier-Stokes equations show t...

Section: Unsteady Rotational Advection-diffusionmentioning

confidence: 99%

mentioning

confidence: 99%

On hybrid residual distribution–Galerkin discretizations for steady and time dependent viscous laminar flows

Dobeš

Ricchiuto

Computer Methods in Applied Mechanics and Engineering

et al. 2015

“…A different approach [22,24] is based on the idea that the steady advection-diffusion equation, Eq. (12), is equivalent to a hyperbolic relaxation system at the steady state.…”

Section: Extension To Viscous Termsmentioning

confidence: 99%

High order residual distribution scheme for Navier-Stokes equations

20th AIAA Computational Fluid Dynamics Conference

Santis

2011

In this work we describe the use of the Residual Distribution schemes applied to the discretization of conservation laws. In particular, emphasis is put on the construction of a third order accurate scheme. We first recall the properties of a Residual Distribution scheme and we show how to construct a high order scheme for advection problems. Furthermore, we show how to speed up the convergence of the implicit scheme to the steady solution by the means of the Jacobian-free technique. We then extend the scheme to the case of advection-diffusion problems. In particular, we propose a new approach in which the residuals of the advection and diffusion terms are distributed together to get high order accuracy. Due to the continuous approximation of the solution, the gradients of the variables are reconstructed at the nodes and then interpolated on the elements. The numerical scheme is used to discretize the advection-diffusion scalar problem and the compressible Navier-Stokes equations.

“…An interesting contribution on the approximation of viscous problem is the work of H. Nishikawa [26,27]. Last, and up to our knowledge the first contribution on higher than second order accurate RD scheme is due to Caraeni [12,13], as well as early work on unsteady and viscous problem.…”

Section: Introductionmentioning

confidence: 99%

A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art

2012

Commun. comput. phys.

Abstract. We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems. We also draw some research perspectives.