Courant condition

One may obtain eqn (5.61a or b) as the Euler equation in the variation of without imposing any prescribed boundary conditions on >j(r), the change or variation of ip r). This is accomplished by introducing the natural boundary conditions (Courant and Hilbert 1953). The necessary condition for to be stationary is that its first variation as given in eqn (5.59)... 

To reiterate, the development of these relations, (2.1)-(2.3), expresses conservation of mass, momentum, and energy across a planar shock discontinuity between an initial and a final uniform state. They are frequently called the jump conditions" because the initial values jump to the final values as the idealized shock wave passes by. It should be pointed out that the assumption of a discontinuity was not required to derive them. They are equally valid for any steady compression wave, connecting two uniform states, whose profile does not change with time. It is important to note that the initial and final states achieved through the shock transition must be states of mechanical equilibrium for these relations to be valid. The time required to reach such equilibrium is arbitrary, providing the transition wave is steady. For a more rigorous discussion of steady compression waves, see Courant and Friedrichs (1948). 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

Often this stability condition is named the Courant condition because it has been proved for the first time by R. Courant, C. Friedrichs and G. Levy in 1928. 

These equations are integrated from some initial conditions. For a specified value of s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and b depend just on x and y (and not u), and the equation is linear if a, b, and/all depend on x and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks. Courant and Hilbert (1953, 1962) Rhee, H. K., R. Aris, and N. R. Amundson, First-Order Partial Differential Equations, vol. I, Theory and Applications of Single Equations, Prentice-Hall, Englewood Cliffs, N.J. (1986) and LeVeque (1992), ibid. 

Second method consists of a straightforward discretization method first order (Euler) explicit in time and finite differences in space. Both the time step and the grid size are kept constant and satisfying the Courant Friedrichs Lewy (CFL) condition to ensure the stability of the calculations. To deal with the transport part we have considered the minmod slope limiting method based on the first order upwind flux and the higher order Richtmyer scheme (see, e.g. Quarteroni and Valli, 1994, Chapter 14). We call this method SlopeLimit. 

Detonation, Strong and Weak. This subject is discussed by Evans 8t Ablow (Ref 2, pp 141-42), but prior to this it. is necessary to discuss the existence and uniqueness of classes of reaction waves for specific boundary conditions , as given in the book of Courant 8c Friedrichs (Ref 1, pp 215-22) and in Ref 2... 

We note that the Jacobi identity for the bracket A, B =(AX, LBX) is not needed for the manifold. tiA to be invariant and for (50) evaluated on it to become equivalent to (55). The skew symmetry of L suffices to guarantee both these properties. If however we begin with the time evolution generated by (50) and define the manifold M as the manifold on which (51) equals zero then the Jacobi identity is the integrability condition for (see Courant (1989)). 

Conditions 10.86 to 10.88 are frequently called the Courant-Friedrichs- Lewy (CFL) convergence conditions [58] and a is called the Courant number. 

A further condition applies to the value of the Courant number, which must exceed 1 with the forward-backward scheme. The requirements that both t and h are positive and that fli and 2 are larger than unity are often difficult to satisfy. Furthermore, if T and h are small, the computation time may become excessive. Similar results are obtained with the other schemes. 

Courant number, used in the Courant-Friedrichs-Lewy necessary stability condition for hyperbolic equations... 

F. Zheng, Z. Chen, and J. Zhang, A finite-difference time-domain method without the Courant stability conditions, IEEE Microw. Guided Wave Lett., vol. 9, no. 11, pp. 441-443, Nov. 1999.doi 10.1109/75.808026... 

A practical limitation of the Eulerian schemes is that the convergence of the numerical integration is achieved only if the adopted timestep At < CAx/u, where the constant C is generally equal to unity. This criterion, called the Courant-Fredrichs-Lewy (CFL) condition, imposes timesteps that are generally much smaller than the physical timescales of interest. 

A point to be noted is that the selection of the number of cells and hence the cell length. Ax, cannot be totally free in any finite difference scheme. The Courant condition suggests that the time integration should not attempt to calculate beyond the spatial domain of influence by using a temperature at a distance beyond the range of influence determined by the characteristic velocity of temperature propagation. Hence... 

The draisity profile realized in actuality is that which minimizes the integral in Equation 1.43. Since and n are functions of z, the calculus of variations must be invoked (Courant and Hilbert, 1953). The resulting condition which must be satisfied throughout the intrafacial region is given by... 

The explicit integration methods, such as leapfrog, prediction-correction or Runge-Kutta methods, are usually used to integrate SPH equations for fluid flows. The explicit time integration is conditionally stable. The time step should satisfy the convective stabihty constraint, i.e., the so-caUed Courant-Friedrichs-Lewy (CFL) condition,... 

For stability, the Courant-Freidrich-Lewy condition has been found to be sufficient... 

For a finite-difference approximation of Eq. (6) a two-step Chodov-Roslyakov second order scheme was used. The oscillations near the difference analogs of the SW and the CS were suppressed by a third-order operator. The time step was chosen to satisfy the Courant-Friedrichs-Levy stability condition. Along the body surface, 70 - 500 cells of the computation grid were situated and along the coordinate r], from 30 to 290 cells were situated. The outer boundary was 25 - 50 Rj from the body. Neither the SW nor the CS were fitted in the computation field that is, the shock capturing algorithm was used. The initial field is as presented in Fig. la. 

Enough artificial or real viscosity must be used to smear the shock waves over at least three cells. The PIC form with a constant of 2.0 is useful for many problems because of its scaling as a function of particle velocity. The time step is often determined by the Courant condition however, for most problems the time step can be estimated in microseconds as... 

Euler s differential equations in conjunction with the introduction of La-grangian multipliers constitute the necessary conditions for a minimum, see Courant and Hilbert [56] or Denn [62]. Thereby the integrand Uq of Eq. (6.29) is extended by the product of appropriate parameters known as Lagrangian multipliers and integrands of the side conditions. In the vectorial representation to be given here, this results in Uq + with the vector of Lagrangian multipliers A and respective vector of integrands < from Eqs. (6.30). To obtain Euler s differential equations, the variation of this expression is equated to zero ... 

In Sect. 3 of their paper [ 132] the authors scrutinize the justification of the use of zero-flux surfaces to define atomic basins from a subsystem variational principle. This contention is based on Hilbert and Courant s generalisation of variation calculus to the case of variable domains. A number of conditions have to be obeyed for this generalisation to be applicable. The authors claimed that one such condition is in general violated, by means of a coim-terexample. Of course, one counterexample suffices but I show here that the calculation in their counterexample is flawed. 

An FDTD method is used to solve Maxwell s equations. To insure stability, the time step must satisfy the Courant stabilify condition ... 

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