# Effects of the Jacobian evaluation on Newton's solution of the Euler equations

2005-09-20
Onur, O
Eyi, Sinan
Newton's method is developed for solving the 2-D Euler equations. The Euler equations are discretized using a finite-volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the finite-difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher-order discretizations on the performance of the solver are analysed. Copyright (c) 2005 John Wiley & Sons, Ltd.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS

# Suggestions

 Two-level finite element method with a stabilizing subgrid for the incompressible Navier-Stokes equations NESLİTÜRK, ALİ İHSAN; Aydın Bayram, Selma; Tezer, Münevver (Wiley, 2008-10-20) We consider the Galerkin finite element method for the incompressible Navier-Stokes equations in two dimensions. The domain is discretized into a set of regular triangular elements and the finite-dimensional spaces emploved consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the Solution, a two-level finite element method with a stabilizing subgrid of a single node is described, and its application to the Navier-Stokes equation is ...
 Time-domain BEM solution of convection-diffusion-type MHD equations Bozkaya, N.; Tezer, Münevver (Wiley, 2008-04-20) The two-dimensional convection-diffusion-type equations are solved by using the boundary element method (BEM) based on the time-dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady-state it...
 Improvements to compressible Euler methods for low-Mach number flows Sabanca, M; Brenner, G; Alemdaroglu, N (Wiley, 2000-09-30) In the present study improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver [R...
 Solution to transient Navier-Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method Bozkaya, Canan; Tezer, Münevver (Wiley, 2009-01-20) The two-dimensional time-dependent Navier-Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equati...
 Effects of the Jacobian evaluation on Newton's solution of the Euler equations Onur, O.; Eyi, Sinan (null; 2005-12-01) A Newton's method is developed for solving the 2-D Euler equations. The Euler equations are discretized using a finite-volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accur...
Citation Formats
O. Onur and S. Eyi, “Effects of the Jacobian evaluation on Newton’s solution of the Euler equations,” INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, pp. 211–231, 2005, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/43593.