Hyperbolic Partial Differential Equation - Hyperbolic System and Conservation Laws

Hyperbolic System and Conservation Laws

There is a connection between a hyperbolic system and a conservation law. Consider a hyperbolic system of one partial differential equation for one unknown function . Then the system has the form

(**) \quad \frac{\partial u}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} {f^j} (u) = 0,

Now can be some quantity with a flux . To show that this quantity is conserved, integrate over a domain

If and are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and to get a conservation law for the quantity in the general form

which means that the time rate of change of in the domain is equal to the net flux of through its boundary . Since this is an equality, it can be concluded that is conserved within .

Read more about this topic:  Hyperbolic Partial Differential Equation

Famous quotes containing the words system, conservation and/or laws:

    “We recognize caste in dogs because we rank ourselves by the familiar dog system, a ladderlike social arrangement wherein one individual outranks all others, the next outranks all but the first, and so on down the hierarchy. But the cat system is more like a wheel, with a high-ranking cat at the hub and the others arranged around the rim, all reluctantly acknowledging the superiority of the despot but not necessarily measuring themselves against one another.”
    —Elizabeth Marshall Thomas. “Strong and Sensitive Cats,” Atlantic Monthly (July 1994)

    “The putting into force of laws which shall secure the conservation of our resources, as far as they may be within the jurisdiction of the Federal Government, including the more important work of saving and restoring our forests and the great improvement of waterways, are all proper government functions which must involve large expenditure if properly performed.”
    —William Howard Taft (1857–1930)

    “The most mighty of nature’s laws is this, that out of Death she brings Life.”
    —Herman Melville (1819–1891)