page_title: Total variation diminishing
section_title: Discretisation
hierarchical_section_title: Total variation diminishing / Discretisation
caption_attribution_description: English: The control volume with the velocity at the faces, nodes and the distance between them
context_page_description: In numerical methods, total variation diminishing is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten.
context_section_description: Consider the steady state one-dimensional convection diffusion equation,
  ,
  where  is the density,  is the velocity vector,  is the property being transported,  is the coefficient of diffusion and  is the source term responsible for generation of the property .
  Making the flux balance of this property about a control volume we get,
    
  Here  is the normal to the surface of control volume.
  Ignoring the source term, the equation further reduces to:
  Assuming 
    and  
  The equation reduces to
  Say,
  From the figure:
  The equation becomes,
  Also the continuity equation has to be satisfied in one of its equivalent forms for this problem:
  Assuming diffusivity is a homogeneous property and equal grid spacing we can say
  we get
  The equation further reduces to
  The equation above can be written as
  where  is the Péclet number
caption: A picture showing the control volume with velocities at the faces,nodes and the distance between them, where 'P' is the node at the center.
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