Conservation form or Eulerian form refers to an arrangement of an equation or system of equations, usually representing an hyperbolic system, that emphasizes that a property represented is conserved, i.e. a type of continuity equation. The term is usually used in the context of continuum mechanics.
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General form
Equations in conservation form take the form
for any conserved quantity
using the divergence theorem. The integral equation states that the change rate of the integral of the quantity
The integral form of such equations is usually the physically more natural formulation, and the differential equation arises from differentiation. Since the integral equation can also have non-differentiable solutions, the equality of both formulations can break down in some cases, leading to weak solutions and severe numerical difficulties in simulations of such equations.
Example
An example of a set of equations written in conservation form are the Euler equations of fluid flow:
Each of these represents the conservation of mass, momentum and energy, respectively.