A one-equation turbulence model for aerodynamic flows

Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to determine the steady transonic flow past an airfoil using an O mesh. Convergence to a steady state is accelerated by the use of a variable time step determined by the local Courant member, and the introduction of a forcing term proportional to the difference between the local total enthalpy and its free stream value.

Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes

This Key Note presents a summary of the development of the Finite Element Method in the field of Electromagnet ic Engineering, together with a description of several contributions of the authors to the Finite Element Method and its application to the solution of electromagnetic problems. First, a self-adaptive mesh scheme is presented in the context of the quasi-static and full-wave analysis of general anisotropic multiconductor arbitrary shaped waveguiding structures. A comparison between two a posteriori error estimates is done. The first one is based on the complete residual of the differential equations defining the problem. The second one is based on a recovery or smoothing technique of the electromagnetic field. Next, an implementation of the first family of Nedelec's curl-conforming elements done by the authors is outlined. Its features are highlighted and compared with other curl-conforming elements. A presentation of an iterative procedure using a numerically exact radiation condition for the analysis of open (scattering and radiation) problems follows. Other contributions of the authors, like the use of wavelet like basis functions and an implementation of a Time Domain Finite Element Method in the context of two-dimensional scattering problems are only mentioned due to the lack of space.

The finite element method in electromagnetics

A new set of benchmarks has been developed for the performance evaluation of highly parallel supercom puters. These consist of five "parallel kernel" bench marks and three "simulated application" benchmarks. Together they mimic the computation and data move ment characteristics of large-scale computational fluid dynamics applications. The principal distinguishing feature of these benchmarks is their "pencil and paper" specification-all details of these benchmarks are specified only algorithmically. In this way many of the difficulties associated with conventional bench- marking approaches on highly parallel systems are avoided.

/pdf/the-nas-parallel-benchmarks-1v48ldcocu.pdf

The Nas Parallel Benchmarks

Modifications to explicit finite difference schemes for solving the shallow water equations for meteorological applications by increasing the time step for the fast gravity waves are analyzed. Terms associated with the gravity waves in the shallow water equations are treated on a coarser grid than those associated with the slow Rossby waves, which contain much more of the available energy and must be treated with higher accuracy, enabling a several-fold increase in time step without degrading the accuracy of the solution. The method is presented in Cartesian and spherical coordinates for a rotating earth, using generalized leapfrog, frozen coefficient, and Fourier filtering finite difference schemes. Computational results verify the numerical stability of the approach.

Explicit large time-step schemes for the shallow water equations

A method of boundary equations for unsteady hyperbolic problems in 3D

In many numerical procedures one wishes to improve the basic approach either to improve efficiency or else to improve accuracy. Frequently this is based on an analysis of the properties of the discrete system being solved. Using a linear algebra approach one then improves the algorithm. We review methods that instead use a continuous analysis and properties of the differential equation rather than the algebraic system. We shall see that frequently one wishes to develop methods that destroy the physical significance of intermediate results. We present cases where this procedure works and others where it fails. Finally we present the opposite case where the physical intuition can be used to develop improved algorithms.

Numerical Methods and Nature

We consider high order methods for the one-dimensional Helmholtz equa- tion and frequency-Maxwell system. We demand that the scheme be higher order even when the coefficients are discontinuous. We discuss the connection between schemes for the second-order scalar Helmholtz equation and the first-order system for the elec- tromagnetic or acoustic applications. AMS subject classifications: 65N06, 78A48, 78M20

/pdf/fourth-order-schemes-for-time-harmonic-wave-equations-with-xt0o30clrf.pdf

Fourth Order Schemes for Time-Harmonic Wave Equations with Discontinuous Coefficients

https://hal.archives-ouvertes.fr/hal-00511298/document

Eli Turkel

Papers

Explicit large time-step schemes for the shallow water equations

A method of boundary equations for unsteady hyperbolic problems in 3D

Numerical Methods and Nature

Fourth Order Schemes for Time-Harmonic Wave Equations with Discontinuous Coefficients

Time Reversed Absorbing Conditions