Multidimensional Upwind Residual Distribution Schemes Research Articles

A Parallel Adaptive Newton-Krylov-Schwarz Method for 3D Compressible Inviscid Flow Simulations

A parallel adaptive pseudo transient Newton-Krylov-Schwarz (αΨNKS) method for the solution of compressible flows is presented. Multidimensional upwind residual distribution schemes are used for space discretisation, while an implicit time-marching scheme is employed for the discretisation of the (pseudo)time derivative. The linear system arising from the Newton method applied to the resulting nonlinear system is solved by the means of Krylov iterations with Schwarz-type preconditioners. A scalable and efficient data structure for theαΨNKS procedure is presented. The main computational kernels are considered, and an extensive analysis is reported to compare the Krylov accelerators, the preconditioning techniques. Results, obtained on a distributed memory computer, are presented for 2D and 3D problems of aeronautical interest on unstructured grids.

Second order blended multidimensional upwind residual distribution scheme for steady and unsteady computations

Multidimensional upwind residual distribution (RD) schemes have become an appealing alternative to more widespread finite volume and finite element methods (FEM) for solving compressible fluid flows. The RD approach allows to construct nonlinear second order and non-oscillatory methods at the same time. They are routinely used for steady state calculations of the complex flow problems, e.g., 3D turbulent transonic industrial-type simulations [H. Deconinck, K. Sermeus, R. Abgrall, Status of multidimensional upwind residual distribution schemes and applications in aeronautics, AAIA Paper 2000-2328, AIAA, 2000; K. Sermeus, H. Deconinck, Drag prediction validation of a multi-dimensional upwind solver, CFD-based aircraft drag prediction and reduction, VKI Lecture Series 2003-02, Von Karman Institute for Fluid Dynamics, Chausée do Waterloo 72, B-1640 Rhode Saint Genèse, Belgium, 2003]. Despite its maturity, some problems are still present for the nonlinear schemes developed up to now: namely a poor iterative convergence for the transonic problems and a decrease of accuracy in smooth parts of the flow, caused by a weak L 2 instability [M. Ricchiuto, Construction and analysis of compact residual discretizations for conservation laws on unstructured meshes. Ph.D. Thesis, Université Libre de Bruxelles, Von Karman Institute for Fluid Dynamics, 2005]. We have developed a new formulation of a blended scheme between the second order linear LDA [R. Abgrall, M. Mezine, Residual distribution scheme for steady problems, 33rd Computational Fluid Dynamics course, VKI Lecture Series 2003-05, Von Karman Institute for Fluid Dynamics, Chausée do Waterloo 72, B-1640 Rhode Saint Genèse, Belgium, 2003] scheme and the first order N scheme. The blending coefficient is based on a simple shock capturing operator and it is properly scaled such that second order accuracy is preserved. The approach is extended to unsteady flows problems using consistent formulation of the LDA scheme with the mass matrix [M. Mezine, M. Ricchiuto, R. Abgrall, H. Deconinck, Monotone and stable residual distribution schemes on prismatic space–time elements for unsteady conservation laws, 33rd Computational Fluid Dynamics Course, Von Karman Institute for Fluid Dynamics, 2003; R. Abgrall, M. Mezine, Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems, J. Comput. Phys. 188 (2003) 16–55]. For the time integration, a three point backward scheme is selected for its accuracy and robustness and the shock capturing operator is modified appropriately, to handle moving shocks. We present a numerical solution of several challenging test cases involving the solution of the Euler equations from the subsonic to the hypersonic regime. All the tests shows very good accuracy, robustness and convergence properties.

Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems

The Residual Distribution (RD) schemes are an alternative to standard high order accurate finite volume schemes. They have several advantages: a better accuracy, a much more compact stencil, easy parallelization. However, they face several problems, at least for steady problems which are the only cases considered here. The solution is obtained via an iterative method. The iterative convergence must be good in order to get spatially accurate solutions, as suggested by the few theoretical results available for the RD schemes. In many cases, especially for systems, the iterative convergence is not sufficient to guaranty the theoretical accuracy. In fact, up to our knowledge, the iterative convergence is correct in only two cases: for first order monotone schemes and the (scalar) Struij’s PSI scheme which is a multidimensional upwind scheme. Up to our knowledge, the iterative convergence is poor for systems, except for the blended scheme of Deconinck et al. [Á. Csík, M. Ricchiuto, H. Deconinck, A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws, J. Comput. Phys. 179(2) (2002) 286–312] and Abgrall [R. Abgrall, Toward the ultimate conservative scheme: following the quest, J. Comput. Phys. 167(2) (2001) 277–315] which are also a genuinely multidimensional upwind scheme. A second drawback is that their construction relies, up to now, on a single first order scheme: the N scheme. However, it is known that standard first order finite volume schemes can be rephrased into a Residual Distribution framework. Unfortunately, the standard way of upgrading the order of accuracy to second order leads to very unsatisfactory results but clearly the construction of good schemes based on a wider class of first order schemes would be interesting. In this paper, we analyze these two problems, and show they are linked. We propose a fix and demonstrate its efficiency on several test cases that cover a wide range of applications. Our solution extends considerably the number of working RD schemes.

An entropy fix for multi-dimensional upwind residual distribution schemes

The residual distribution method allows a genuinely multi-dimensional generalization of the flux difference splitting schemes that are commonly used for dimension-by-dimension upwinding in finite volume discretization methods. Despite their many advantages, the residual distribution schemes also share an important deficiency with the Roe-type flux difference splitting schemes: they admit expansion shocks and thus do not respect the entropy condition, i.e. the second law of thermodynamics. In this paper the entropy condition violation of the residual distribution method is analyzed and an entropy fix is presented. By adding the entropy fix the deficiency is cured, as demonstrated by two examples of steady supersonic Euler flow.

Numerical Simulation of the Slider Air Bearing Problem of Hard Disk Drives by Two Multidimensional Upwind Residual Distribution Schemes over Unstructured Triangular Meshes

In this paper we present two multigrid numerical schemes over unstructured triangular meshes that solve the slider air bearing problem of hard disk drives. For each fixed slider attitude, the air bearing pressure is obtained by solving the generalized Reynolds equation. The convection part of the equation is modeled in one scheme by the PSI multidimensional upwind residual distribution approach and in the other scheme by the SUPG finite element approach cast in residual distribution form. In both schemes, a linear Galerkin method is used to discretize the diffusion terms. In addition, a non-nested multigrid iteration technique is used to speed up the convergence rate. Finally, the balanced steady state flying attitude of the slider subject to pre-applied suspension force and torques is obtained by a Quasi–Newton iteration method (Broyden's method), and the results of the numerical solutions are compared to each other and to experimental data.

Monotone Residual Distribution Schemes for the Ideal Magnetohydrodynamic Equations on Unstructured Grids

Multidimensional upwind residual distribution schemes are applied to the eight-wave equations of ideal magnetohydrodynamics. These schemes extend the high-resolution upwind finite volume methodology to a truly multidimensional finite element context on unstructured grids. Both first- and second-order linear and second-order nonlinear monotonicity preserving schemes are discussed. An approximate conservative linearization technique is derived for general hyperbolic systems together with a conservative correction technique, which guarantees the full conservation of the convective fluxes. The solenoidal condition of the magnetic field is enforced by Powell's source term approach. Both implicit and explicit time-integration strategies have been implemented. The spatial accuracy and the shock-capturing properties of the schemes in the steady state are investigated numerically. Computational results are presented for a bow shock over a cylinder and for a supermagnetosonic flow in a channel