First-Order, Quasi-linear Equations and Method of Characteristics

2012 ◽  
pp. 201-226
Author(s):  
Lokenath Debnath
Keyword(s):  
Method Of Characteristics ◽  
Linear Equations ◽  
First Order

Partially Invariant Solutions for Two-dimensional Ideal MHD Equations

1990 ◽  
Vol 45 (11-12) ◽  
pp. 1219-1229 ◽  
Author(s):  
D.-A. Becker ◽  
E. W. Richter
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Mhd Equations ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
First Order ◽  
Partially Invariant Solutions ◽  
Quasilinear Systems ◽  
Ideal Mhd ◽  
Non Linear Equations

AbstractA generalization of the usual method of similarity analysis of differential equations, the method of partially invariant solutions, was introduced by Ovsiannikov. The degree of non-invariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of first-order partial differential equations. We apply the algorithm to the non-linear equations of the two-dimensional non-stationary ideal MHD with a magnetic field perpendicular to the plane of motion.

A Comparative Study of Numerical Techniques for Nonequilibrium Method of Characteristics

1992 ◽  
Author(s):  
T. Gary Yip ◽  
David M. Crook ◽  
Timothy P. Buell
Keyword(s):  
Method Of Characteristics ◽  
Solution Process ◽  
Initial Pressure ◽  
Single Step ◽  
Supersonic Combustion ◽  
Duct Flow ◽  
Unit Process ◽  
First Order ◽  
Constant Area ◽  
Fifth Order

Abstract Three techniques which employ different approaches for obtaining a method of characteristics solution for chemical non-equilibrium flows are reviewed and compared. Two features of the solution process are evaluated to determine their effect on the accuracy of the solution. The first aspect to be considered is the integration of the stiff conservation equations in a unit process. A new fifth-order accurate, multi-step integration routine is contrasted with a first-order accurate, single-step forward differencing scheme. The second comparison is designed to determine if a solution of the flowfield along continuous streamlines is superior to one along discontinuous segments of the streamlines. Tests are performed, using a chemical model describing the supersonic combustion of H2-air. Calculations of single unit processes are used to validate the techniques and to determine suitable grid sizes. Solutions for constant area duct flow show that the techniques which use the multi-step integration routine are more accurate. Results from the constant area duct test, for an initial pressure of 3.685 atm, show that a method of characteristics technique which utilizes continuous streamlines is able to converge at a grid size two orders of magnitude larger than that needed by a technique which uses discontinuous segments of streamlines.

First Order Linear Equations

2001 ◽  
pp. 51-79
Author(s):  
Martin Bohner ◽  
Allan Peterson
Keyword(s):  
Linear Equations ◽  
Order Linear ◽  
First Order

scholarly journals Minimization of Deviations of Gear Real Tooth Surfaces Determined by Coordinate Measurements

1993 ◽  
Vol 115 (4) ◽  
pp. 995-1001 ◽  
Author(s):  
F. L. Litvin ◽  
C. Kuan ◽  
J. C. Wang ◽  
R. F. Handschuh ◽  
J. Masseth ◽  
...  
Keyword(s):  
Linear Equations ◽  
Least Square Method ◽  
Least Square ◽  
Tooth Surface ◽  
Hypoid Gear ◽  
Overdetermined System ◽  
Entire Surface ◽  
First Order ◽  
Tooth Surfaces ◽  
Coordinate Measurements

The deviations of a gear’s real tooth surface from the theoretical surface are determined by coordinate measurements at the grid of the surface. A method has been developed to transform the deviations from Cartesian coordinates to those along the normal at the measurement locations. Equations are derived that relate the first order deviations with the adjustment to the manufacturing machine tool settings. The deviations of the entire surface are minimized. The minimization is achieved by application of the least-square method for an overdetermined system of linear equations. The proposed method is illustrated with a numerical example for hypoid gear and pinion.

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