WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions

The paper extends weighted essentially non-oscillatory (WENO) methods to three dimensional mixed-element unstructured meshes, comprising tetrahedral, hexahedral, prismatic and pyramidal elements. Numerical results illustrate the convergence rates and non-oscillatory properties of the schemes for various smooth and discontinuous solutions test cases and the compressible Euler equations on various types of grids. Schemes of up to fifth order of spatial accuracy are considered

On the use of TVD fluxes in ENO and WENO schemes

Very high order methods, such as ENO/WENO methods [21, 30, 19], Runge-Kutta Discontinuous Galerkin Finite Element Methods [12] and ADER methods [54,46] often use high order (e.g. fifth order) polynomial reconstruction of the solution of a lower (first) order monotone flux as the building block. In this paper we propose to use second order TVD fluxes in the framework of such methods and apply the principle to the finite-volume ENO, WENO and MPWENO schemes. We call the new improved schemes the ENO-TVD, WENO-TVD and MPWENO-TVD schemes respectively. They include both upwind and centred schemes with non-stuggered meshes. Numerical results suggest that our schemes are superior to original schemes with first order fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features

Very high-order methods for 3D arbitrary unstructured grids

Understanding the motion of fluids is crucial for the development and analysis of new designs and processes in science and engineering. Unstructured meshes are used in this context since they allow the analysis of the behaviour of complicated geometries and configurations that characterise the designs of engineering structures today. The existing numerical methods developed for unstructured meshes suffer from poor computational efficiency, and their applicability is not universal for any type of unstructured meshes. High-resolution high-order accurate numerical methods are required for obtaining a reasonable guarantee of physically meaningful results and to be able to accurately resolve complicated flow phenomena that occur in a number of processes, such as resolving turbulent flows, for direct numerical simulation of Navier-Stokes equations, acoustics etc. The aim of this research project is to establish and implement universal, high-resolution, very high-order, non-oscillatory finite-volume methods for 3D unstructured meshes. A new class of linear and WENO schemes of very high-order of accuracy (5 th ) has been developed. The key element of this approach is a high-order reconstruction process that can be applied to any type of meshes. The linear schemes which are suited for problems with smooth solutions, employ a single reconstruction polynomial obtained from a close spatial proximity. In the WENO schemes the reconstruction polynomials, arising from different topological regions, are non-linearly combined to provide high-order of accuracy and shock capturing features. The performance of the developed schemes in terms of accuracy, non-oscillatory behaviour and flexibility to handle any type of 3D unstructured meshes has been assessed in a series of test problems. The linear and WENO schemes presented achieve very high-order of accuracy (5 th ). This is the first class of WENO schemes in the finite volume context that possess highorder of accuracy and robust non-oscillatory behaviour for any type of unstructured meshes. The schemes have been employed in a newly developed 3D unstructured solver (UCNS3D). UCNS3D utilises unstructured grids consisted of tetrahedrals, pyramids, prisms and hexahedral elements and has been parallelised using the MPI framework. The high parallel efficiency achieved enables the large scale computations required for the analysis of new designs and processes in science and engineering.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

UCNS3D: An open-source high-order finite-volume unstructured CFD solver

UCNS3D is an open-source computational solver for compressible flows on unstructured meshes. State-of-the-art high-order methods and their associated benefits can now be implemented for industrial-scale CFD problems due to the flexibility and highly-automated generation offered by unstructured meshes. We present the governing equations of the physical models employed in UCNS3D, and the numerical framework developed for their solution. The code has been designed so that extended to other systems of equations and numerical models is straightforward. The employed methods are validated towards a series of stringent well-established test problems against experimental or analytical solutions, where the full capabilities of UCNS3D in terms of applications spectrum, robustness, efficiency, and accuracy are demonstrated.European Union funding: 314139, 653838 and 823767. Engineering and Physical Sciences Research Council (EPSRC): EP/L000261/1, EP/P020259/1, EP/G069581/1, EP/T518104/1 and 13794 Innovate UK: 26326

Conservative numerical methods for model kinetic equations

A new conservative discrete ordinate method for nonlinear model kinetic equations is proposed. The conservation property with respect to the collision integral is achieved by satisfying at the discrete level approximation conditions used in deriving the model collision integrals. Additionally to the conservation property, the method ensures the correct approximation of the heat fluxes. Numerical examples of flows with large gradients are provided for the Shakhov and Rykov model kinetic equations

Derivative Riemann problem and ADER schemes

Not availabl

Uniformly high-order schemes on arbitrary unstructured meshes for advection diffusion

The paper presents a linear high-order method for advection-di®usion conser- vation laws on three dimensional mixed-element unstructured meshes. The key ingredient of the method is a reconstruction procedure in local compu- tational coordinates. Numerical results illustrate the convergence rates for the linear equation and a non-linear hyperbolic system with di®usion terms for various types of meshes

Exact and approximate solutions of Riemann problems in non-linear elasticity

Eulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to non-linear elastic theory. In this paper one such class of solver is examined based upon characteristic tracing in conjunction with high-order monotonicity preserving weighted essentially non-oscillatory (MPWENO) reconstruction. Furthermore, a new iterative method for finding exact solutions of the Riemann problem in non-linear elasticity is presented. Access to exact solutions enables an assessment of the performance of the numerical techniques with focus on the resolution of the seven wave structure. The governing model represents a special case of a more general theory describing additional physics such as material plasticity. The numerical scheme therefore provides a firm basis for extension to simulate more complex physical phenomena. Comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations is presented

Planar Gas Expansion under Intensive Nanosecond Laser Evaporation into Vacuum as Applied to Time-of-Flight Analysis

A computational investigation of the dynamics of gas expansion due to intense nanosecond laser evaporation into vacuum has been carried out. The problem is solved in a one-dimensional approximation, which simplifies calculations and at the same time allows one to analyze the main features of the expansion dynamics. For analysis we use three different approaches. Two of them are based on kinetic analysis via the direct simulation Monte Carlo (DSMC) method and numerical solution of the model Bhatnagar–Gross–Krook (BGK) equation. The third one focuses on derivation of an analytical continuum solution. Emphasis is placed on the analysis of the velocity distribution function and the average energy of particles passing through the time-of-flight detector on the normal to the evaporation surface, which is important for interpreting experimental measurements. The formulated problem is quite difficult as the considered flow is time-dependent, contains discontinuities in boundary conditions and involves large variations of local Knudsen numbers as well as steep gradients of the velocity distribution function. Data were obtained on the particle energy in the time-of-flight distribution for the range of regimes from the free molecular flow to continuum one. The maximum attainable average energy of particles in the time-of-flight distribution is determined. The non-monotonicity of the energy increase was found, which is explained based on analysis of the velocity distribution of particles