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
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
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