Many physical problems can be cast in the form of conservation law PDEs. For simplicity, we begin by discussing conservation law PDEs involving only one space variable.
Suppose that some material quantity M, such as fluid or heat, moves parallel to the x-axis and that the flow field is independent of y and z. Under this assumption of one dimensional flow, it is sufficient to examine the flow in a cylinder of cross-sectional area A whose generators are parallel to the x-axis. Let be the linear density of material quantity M (amount of M per unit length per unit area) and let denote the flux of material M in the x-direction. The flux is the amount of material M that crosses a unit planar area perpendicular to the direction of flow per unit time.
Consider flow of M through the portion of the cylinder defined by a < x < b. If there are no additions (sources) or removals (sinks) of material M from within the interval , then the following conservation principle holds.
The time rate of change of the amount of material M in the interval equals the flux of material M into at x=a minus the flux of material M out of at x=b.