Reconstruction vs Interpolation

Some words here about the fondamental difference between reconstruction and interpolation.

Definition :

  • Interpolation builds a continuous function of x given known discrete point values of a function
  • Reconstruction builds a continuous function of x given known discrete mean values of a function

In a Finite Volume framework, physical fields are stored inside control volumes (the cells) and PDEs are usualy solved in conserative form. In fact, we are using mean values of the continuous field inside the control volumes, and not the value at the center of the cell.
Hence, reconstruction techniques should be used and not interpolation.

At the center a the cell, second order reconstruction schemes and second order interpolation schemes are equal ; this is why we usualy use regular inteprolation methods and finite difference schemes.
However, when searching for higher order schemes, it is necessary to use appropriate reconstruction (and their derivatives).

Error estimation should take care of this fact as, when comparing to reference values, mean values (or point values) should be used in a coherent manner.
For example, if, for verification purpose, one provides exact point values solution for a temperature field and compares it to FV’s mean values of the field, one can never expect better than 2nd order spatial convergence.

Notus provides several base reconstruction and interpolation functions (as well as finite differences schemes) that can be used. More information can be found in the documentation :
Developer corner (see Discretization/Node level schemes)

PS : conversly, passing from point based values to mean values can be done via appropriate classical quadrature techniques (also available in Notus) based on interpolation functions of discrete values.