A Finite Difference Scheme and an Existence Theorem for a Nonlinear Hyperbolic System of Differential Equations

1971 ◽  
Vol 8 (3) ◽  
pp. 524-535 ◽  
Author(s):  
Pierre Jamet
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Scheme ◽  
Existence Theorem ◽  
Hyperbolic System ◽  
Finite Difference Scheme ◽  
System Of Differential Equations ◽  
Nonlinear Hyperbolic System

scholarly journals THE EXACT FINITE-DIFFERENCE SCHEME FOR VECTOR BOUNDARY‐VALUE PROBLEMS WITH PIECE‐WISE CONSTANT COEFFICIENTS

1998 ◽  
Vol 3 (1) ◽  
pp. 114-123
Author(s):  
H. Kalis
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Boundary Value ◽  
Parabolic Type ◽  
Second Order ◽  
Mhd Equations ◽  
System Of Differential Equations ◽  
Constant Coefficients

We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schemes with perturbation coefficient of function‐matrix for solving the system of differential equations. Special finite‐difference approximations are constructed for a steady‐state boundary‐value problem, systems of parabolic type partial differential equations, a system of two MHD equations, 2‐D flows and MHD‐flows equations in curvilinear orthogonal coordinates.

scholarly journals On a hyperbolic system arising in liquid crystals modeling

2018 ◽  
Vol 15 (01) ◽  
pp. 15-35 ◽  
Author(s):  
Eduard Feireisl ◽  
Elisabetta Rocca ◽  
Giulio Schimperna ◽  
Arghir Zarnescu
Keyword(s):  
Liquid Crystals ◽  
Initial Data ◽  
Existence Theorem ◽  
Hyperbolic System ◽  
Finite Energy ◽  
Classical Solutions ◽  
System Of Differential Equations ◽  
Dissipative Solutions ◽  
Nonlinear Hyperbolic System ◽  
Time Existence

We consider a model of liquid crystals, based on a nonlinear hyperbolic system of differential equations, that represents an inviscid version of the model proposed by Qian and Sheng. A new concept of dissipative solution is proposed, for which a global-in-time existence theorem is shown. The dissipative solutions enjoy the following properties: (i) they exist globally in time for any finite energy initial data; (ii) dissipative solutions enjoying certain smoothness are classical solutions; (iii) a dissipative solution coincides with a strong solution originating from the same initial data as long as the latter exists.

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