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Second-order phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy. [6]
Landau theory (also known as Ginzburg–Landau theory, despite the confusing name [1]) in physics is a theory that Lev Landau introduced in an attempt to formulate a general theory of continuous (i.e., second-order) phase transitions. [2]
The glass transition presents features of a second-order transition since thermal studies often indicate that the molar Gibbs energies, molar enthalpies, and the molar volumes of the two phases, i.e., the melt and the glass, are equal, while the heat capacity and the expansivity are discontinuous.
In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix. Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. [1] [2]
One easily sees that those discontinuities are all removable. By the first paragraph, there does not exist a function that is continuous at every rational point, but discontinuous at every irrational point. The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere. These discontinuities are all ...
The second project was the development of discontinuous Galerkin (DG) methods for diffusion operators. It started with the discovery of the recovery method for representing the 1D diffusion operator. Starting in 2004, the recovery-based DG (RDG) [ 31 ] has been shown an accuracy of the order 3p+1 or 3p+2 for even or odd polynomial-space degree p.
The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward). For nonlinear equations, this procedure provides the best results. For linear equations, the MacCormack scheme is equivalent to the Lax–Wendroff method .
By standard changes of variables, a second order equation can often be changed to one of the form d 2 w d z 2 = f ( z ) w {\displaystyle {\frac {d^{2}w}{dz^{2}}}=f(z)w} where f is holomorphic in a simply-connected region and w is a solution of the differential equation.