High-order Spatial Discretization Schemes Research Articles

Convergence towards High-Speed Steady States Using High-Order Accurate Shock-Capturing Schemes

Creating time-marching unsteady governing equations for a steady state in high-speed flows is not a trivial task. Residue convergence in time cannot be achieved when using most low- and high-order spatial discretization schemes. Recently, high-order, weighted, essentially non-oscillatory schemes have been specially designed for steady-state simulations. They have been shown to be capable of achieving machine precision residues when simulating the Euler equations under canonical coordinates. In the present work, we review these schemes and show that they can also achieve machine residues when simulating the Navier–Stokes equations under generalized coordinates. This is carried out by considering three supersonic flows of perfect fluids, namely the flow upstream a cylinder, the flow over a blunt wedge, and the flow over a compression ramp.

On the properties of high-order least-squares finite-volume schemes

High-order finite-volume schemes based on polynomial least-squares methods are an active research topic for the discretization of hyperbolic equations as they allow to obtain high-order spatial discretization schemes in arbitrary meshes. However, few studies have analyzed their performance in good-quality/near-to-uniform meshes, which are commonly used as a meshing strategy in zones where turbulent effects are important. In this paper, the theoretical numerical properties of commonly used least-squares (LSQ) k-exact high-order finite volume schemes are studied in one-dimensional and in several two-dimensional meshes (with some remarks regarding their properties in three-dimensional meshes). These results are compared to those obtained using fully-constrained polynomial reconstructions only compatible with structured meshes. The numerical properties of the schemes are investigated through the von Neumann analysis methodology applied to the one-dimensional and two-dimensional finite volume formulation, including temporal discretization errors. This analysis is also extended to non-uniform and unstructured two-dimensional meshes. At last, the schemes are tested with several numerical experiments using the linear advection, the Euler equations and the Navier-Stokes equations. The analysis of both studies yields similar conclusions regarding the numerical errors and stability of the different studied schemes showing that the high-order least-squares finite volume schemes yield stable and robust results across different uniform and non-uniform unstructured meshes. However, their performance is heavily degraded in simulations with low mesh resolution compared to schemes specially catered to structured meshes. On the other hand, the latter schemes lack stability and robustness in general structured meshes and its formulation cannot be straightforwardly extended to unstructured meshes. Moreover, this work shows that the use of weighted-LSQ can drastically improve the results of LSQ schemes in under-resolved simulations.

Delta-form-based method of solving high order spatial discretization schemes for neutron transport

Delta-form-based methods for solving high order spatial discretization schemes are introduced into the reactor SN transport equation. Due to the nature of the delta-form, the final numerical accuracy only depends on the residuals on the right side of the discrete equations and have nothing to do with the parts on the left side. Therefore, various high order spatial discretization methods can be easily adopted for only the transport term on the right side of the discrete equations. Then the simplest step or other robust schemes can be adopted to discretize the increment on the left hand side to ensure the good iterative convergence. The delta-form framework makes the sweeping and iterative strategies of various high order spatial discretization methods be completely the same with those of the traditional SN codes, only by adding the residuals into the source terms. In this paper, the flux limiter method and weighted essentially non-oscillatory scheme are used for the verification purpose to only show the advantages of the introduction of delta-form-based solving methods and other high order spatial discretization methods can be also easily extended to solve the SN transport equations. Numerical solutions indicate the correctness and effectiveness of delta-form-based solving method.

A directional ghost-cell immersed boundary method for incompressible flows

This paper describes an efficient ghost-cell immersed boundary method for incompressible flow simulations. The method employs a locally directional extrapolation scheme along all discretization directions for the ghost values. Additionally, it involves fictitious discrete boundary forcing terms instead of the ghost values in the governing equations. In this way, the boundary is represented more accurately than in prior IBMs and it is possible to fulfill the divergence-free condition. When combined with high-order spatial discretization schemes, the IBM order is reduced locally near the immersed boundary in a step-wise manner. In this way, the method delivers more compact stencils and is able to deal with sharp interfaces. By handling the distance from the ghost point to the boundary carefully, the proposed method delivers lower truncation errors than standard IBM, with clean (persistent) convergence rate and enhanced stability. The parallel implementation of this approach is straightforward. Its accuracy has been checked by considering a variety of test cases, including irregular, three-dimensional, and moving boundaries. The local accuracy for the proposed method is formally second order, and is measured to be close to this value using numerical tests.

Seeding Chaos: The Dire Consequences of Numerical Noise in NWP Perturbation Experiments

AbstractPerturbation experiments are a common technique used to study how differences between model simulations evolve within chaotic systems. Such perturbation experiments include modifications to initial conditions (including those involved with data assimilation), boundary conditions, and model parameterizations. We have discovered, however, that any difference between model simulations produces a rapid propagation of very small changes throughout all prognostic model variables at a rate many times the speed of sound. The rapid propagation seems to be due to the model’s higher-order spatial discretization schemes, allowing the communication of numerical error across many grid points with each time step. This phenomenon is found to be unavoidable within the Weather Research and Forecasting (WRF) Model even when using techniques such as digital filtering or numerical diffusion.These small differences quickly spread across the entire model domain. While these errors initially are on the order of a millionth of a degree with respect to temperature, for example, they can grow rapidly through nonlinear chaotic processes where moist processes are occurring. Subsequent evolution can produce within a day significant changes comparable in magnitude to high-impact weather events such as regions of heavy rainfall or the existence of rotating supercells. Most importantly, these unrealistic perturbations can contaminate experimental results, giving the false impression that realistic physical processes play a role. This study characterizes the propagation and growth of this type of noise through chaos, shows examples for various perturbation strategies, and discusses the important implications for past and future studies that are likely affected by this phenomenon.

High-resolution method for evolving complex interface networks

In this paper we describe a high-resolution transport formulation of the regional level-set approach for an improved prediction of the evolution of complex interface networks. The novelty of this method is twofold: (i) construction of local level sets and reconstruction of a global level set, (ii) local transport of the interface network by employing high-order spatial discretization schemes for improved representation of complex topologies. Various numerical test cases of multi-region flow problems, including triple-point advection, single vortex flow, mean curvature flow, normal driven flow, dry foam dynamics and shock–bubble interaction show that the method is accurate and suitable for a wide range of complex interface-network evolutions. Its overall computational cost is comparable to the Semi-Lagrangian regional level-set method while the prediction accuracy is significantly improved. The approach thus offers a viable alternative to previous interface-network level-set method.

High Order Implicit-explicit General Linear Methods with Optimized Stability Regions

In the numerical solution of partial differential equations using a method-of-lines approach, the availability of high order spatial discretization schemes motivates the development of sophisticated high order time integration methods. For multiphysics problems with both stiff and nonstiff terms implicit-explicit (IMEX) time stepping methods attempt to combine the lower cost advantage of explicit schemes with the favorable stability properties of implicit schemes. Existing high order IMEX Runge--Kutta or linear multistep methods, however, suffer from accuracy or stability limitations. This work shows that IMEX general linear methods (GLMs) are competitive alternatives to classic IMEX schemes for large problems arising in practice. High order IMEX-GLMs are constructed in the partitioned GLM framework developed earlier by the authors [J. Sci. Comput., 61 (2014), pp. 119--144]. The stability regions of the new schemes are optimized numerically. The resulting IMEX-GLMs have similar stability properties as IMEX Runge--Kutta methods, but they do not suffer from order reduction and are superior in terms of accuracy and efficiency. The new IMEX-GLMs have considerably better stability properties than the IMEX linear multistep methods. Numerical experiments with two- and three-dimensional test problems illustrate the potential of the new schemes to speed up complex applications.

Gray radiative conductive 2D modeling using discrete ordinates method with multidimensional spatial scheme and non-uniform grid

The problem of radiation heat transfer coupled with other modes of heat transfer is important in many engineering applications. Some examples are the heat transfer in glass fabrication and in thermal isolation materials. The angular and spatial discretization of the radiative transport equation in the discrete ordinates method (DOM) plays an important role to obtain accurate numerical results. Two high order spatial discretization schemes are used and compared. One spatial discretization scheme is unidirectional and the other is multidimensional interpolating scheme. Different angular quadratures are selected and tested to obtain accurate results with less computational time. The radiative heat transport equation is solved using the conventional procedure of solution for DOM and the algorithm is validated by comparison with literature exact solutions for different two-dimensional cases. The radiative source term in the energy equation is computed from intensities field. The radiative conductive model is validated by comparison with test cases solutions from the literature. Non-uniform grids are implemented for multidimensional spatial scheme and the results are compared with the result of uniform grid showing agreement. Also, the non-uniform grids are tested in cases of high temperature gradients. To accelerate convergence an adequate relaxation factors in radiative heat transport equation and in energy equation are used. The method can be used to handle cases with reflecting boundaries.