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A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions (using the squared scalar wave velocity ...
In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form. Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations ...
e. In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic and parabolic partial differential equation.
Godunov's scheme. In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei Godunov in 1959, [ 1] for solving partial differential equations. One can think of this method as a conservative finite volume method which solves exact, or approximate Riemann problems at each inter ...
Overview. The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation. for a system of particles at coordinates . The function is the system's Hamiltonian giving the system's energy. The solution of the equation is the action functional, , [ 4] called Hamilton's principal function in older textbooks.
Miscellanea. v. t. e. The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. [ a] Functionals are often expressed as definite integrals ...
For example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...
In computational physics, the term advection scheme refers to a class of numerical discretization methods for solving hyperbolic partial differential equations. In the so-called upwind schemes typically, the so-called upstream variables are used to calculate the derivatives in a flow field. That is, derivatives are estimated using a set of data ...