The generic scalar transport equation is a general partial differential equation that describes transport phenomena such as heat transfer, mass transfer, fluid dynamics (momentum transfer), etc. A general form of the equation is
where is called the flux, and is called the source.
All the transfer processes express a certain conservation principle. In this respect, any differential equation addresses a certain quantity as its dependent variable and thus expresses the balance between the phenomena affecting the evolution of this quantity. For example, the temperature of a fluid in a heated pipe is affected by convection due to the solid-fluid interface, and due to the fluid-fluid interaction. Furthermore, temperature is also diffused inside the fluid. For a steady-state problem, with the absence of sources, a differential equation governing the temperature will express a balance between convection and diffusion.
A brief inspection of the equations governing various transport phenomena reveal that all of these equations can be put into a generic form thus allowing a systematic approach for a computer simulation. For example, the conservation equation of a concentration of a substance is
where denotes the velocity field, denotes the diffusion flux of the chemical species, and denotes the rate of generation of caused by chemical reaction.
The x-momentum equation for a Newtonian fluid can be written as
where is the body force in the x-direction and includes the viscous terms that are not expressed by
Upon inspection of the above equations, it can be inferred that all the dependent variables seem to obey a generalized conservation principle. If the dependent variable (scalar or vector) is denoted by , the generic differential equation is[1]
where is the diffusion coefficient, or diffusivity.
The objective of all discretization techniques (finite difference, finite element, finite volume, boundary element, etc.) is to devise a mathematical formulation to transform each of these terms into an algebraic equation. Once applied to all control volumes in a given mesh, we obtain a full linear system of equations that needs to be solved.
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Some equations that governs the dynamics of financial derivatives in financial markets can be also categorized as generic scalar transport equations. Examples include the Black-Scholes equation.