Hyperbolic Partial Differential Equations - 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 Equations

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

    “The golden mean in ethics, as in physics, is the centre of the system and that about which all revolve, and though to a distant and plodding planet it be an uttermost extreme, yet one day, when that planet’s year is completed, it will be found to be central.”
    —Henry David Thoreau (1817–1862)

    “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)

    “There is something servile in the habit of seeking after a law which we may obey. We may study the laws of matter at and for our convenience, but a successful life knows no law.”
    —Henry David Thoreau (1817–1862)