Model Solution Library


  CSDMS Model Solution Library

This is a collection of analytic or closed-form solutions to a variety of different mathematical models that are used in the realm of surface process dynamics. It is provided for the purpose of model validation by the CSDMS Cyber-informatics and Numerics Working Group. In the near future, this will include links to pages or papers where the solutions are given and described.

Navier-Stokes Equation

Batchelor vortex (See: Batchelor vortex, approximate solution.)
Burgers vortex (See: Burgers vortex.)
Couette flow (See: Couette flow.)
Hill spherical vortex (See: Vortex ring.)
Lamb-Chaplygin Dipole vortex
Lamb-Oseen vortex (See: Lamb vortex.)
Rankine vortex (See: Rankine vortex. This is a model, not an actual solution.)
Taylor-Couette flow (See: Taylor-Couette flow.)
Taylor-Dean flow

Shallow Water Equations

Inclined plane solution
Dam Break Characteristic Solution
Similarity solutions

Glacier Flow Equations

Halfar (1983) radially-symmetric (Glen Law) similarity solution
Bueler et al. (****) similarity solution

Potential Flow, 2D

Point source or sink
Free vortex (inviscid flow)
Flow around a semi-infinite plate (power-law conformal map: n=1/2)
Flow around a right-angle corner (power-law conformal map: n=2/3)
Uniform flow (power-law conformal map: n = 1)
Power-law conformal map: n=3/2
Flow through a right-angle corner or at a stagnation point (power-law conformal map: n=2)
Flow into a 60-degree corner (power-law conformal map: n=3)
Doublet solution (source-sink pair; power-law conformal map: n=-1)
Quadrupole solution (power-law conformal map: n=-2)
Joukowski airfoil solution
Darcy flow solutions

Jet, Wake and Mixing Layer Solutions

Albertson 2D turbulent jet approximation
Goertler 2D turbulent jet solution
Peckham 2D turbulent jet solution (Peckham, 2008; Goertler and Tollmien are special cases.)
Tollmien 2D turbulent jet solution

Channel and Pipe Flow Solutions

Poisseuille Flow (See: Poiseuille flow.)

Driven Cavity Solutions

Lid-driven or buoyancy-driven, etc. ?
Numeric solution
Analytic solution

Boundary Layer Equation

Blasius (See: Blasius boundary layer.)
Falkners-Skan solution (See: Blasius boundary layer.)
Stokes First Problem
Stokes Second Problem

Stokes Settling Solutions

Water Wave Solutions and Models

Airy waves
Capillary waves
Cnoidal waves
Kelvin waves
Roll waves
Rossby waves
Stokes wave

Ekman Spiral Solution

River Meandering Model

G. Seminara analytic solutions ??

Landscape Evolution Equations

Steady-state, slope vs. area solution
Terry Smith longitudinal profile solutions
Peckham steady-state, uniform-rainrate solutions to "ideal landform equation"

Stratigraphic Evolution Equations

Peckham (2008) prograding solutions (obtained using Laplace transforms, including traveling-wave solutions)

Coastline Evolution Equations

Larson, M., H. Hanson, and N. C. Kraus (1987), Analytical solutions of the one‐line model of shoreline change, Tech. Rep. CERC‐87‐15, U.S. Army Waterw. Exp. Stn., Vicksburg, Miss.

Infiltration Theory

Solutions in: Smith, R.E. (2002) Infiltration Theory for Hydrologic Applications, AGU monograph.

Wave Equation

d'Alembert solution (See: d'Alembert formula.)

Heat Equation

Gaussian, radially-symmetric similarity solution

Laplace Equation

Many, via separation of variables method.

Poisson Equation

Many, from Poisson representation formula

Minimal Surface Equation

Inclined plane solution
Helicoid solution (Meusnier, 1776) See: Helicoid.
Catenoid solution (Meusnier, 1776) See: Catenoid solution and Catenoid.
Scherk's first surface solution (Scherk, 1834) See: Scherk surface.
Scherk's second surface solution (Scherk, 1834)
Costa surface solution (Costa, 1984) See: Costa surface.

Burgers' Equation

Many, via the Cole-Hopf transformation to Heat Equation

Soliton Solutions