More test problems

[ivan-pi]

We should add more test problems to exercise the PDECHEB code in as many modes as possible.

A few relevant codes in this area are:

• PDEONE
• PDECOL
• EPDCOL
• SKMRES
• BACOL (family of solvers)
• NAG Chapter D03 (PDE)
• chebfun

A few more solvers are given in the GAMS Class I2a1a.

Also the following works (and works citing these) may be worth looking into:

• A comparison of Galerkin, collocation and the method of lines for partial differential equations (1978)
• Computational Galerkin Methods (1984)
• A Method for the Spatial Discretization of Parabolic Equations in One Space Variable (1990) (this paper forms the basis of MATLAB's pdepe solver)
• Orthogonal Collocation Revisited (2020)

Lots of parabolic-elliptic initial-value problems can be found in textbooks related to chemical engineering, electrochemistry, transport phenomena, etc.

[ivan-pi]

Two problems can be found in Berzin's early work:

Berzins, M., & Dew, P. M. (1980). A note on the extension of the Chebyshev method to quasi-linear parabolic PDEs with mixed boundary conditions. International Journal of Computer Mathematics, 8(3), 249-263. https://doi.org/10.1080/00207168008803210