# Introduction

Proteus is a Python package for rapidly developing computer models and numerical methods. It is focused on models of continuum mechanical processes described by partial differential equations and on discretizations and solvers for computing approximate solutions to these equations. Proteus consists of a collection of Python modules and scripts. Proteus also uses several C, C++, and Fortran libraries, which are either external open source packages or part of Proteus, and several open source Python packages.

The design of Proteus is organized around two goals:

Make it easy to solve new model equations with existing numerical methods

Make it easy to solve existing model equations with new numerical methods

We want to improve the development process for models *and*
methods. Proteus is not intended to be an expert system for solving
partial differential equations. In fact, effective numerical methods
are often physics-based. Nevertheless many physical models are
mathematically represented by the same small set of differential
operators, and effective numerical methods can be developed with minor
adjustments to existing methods. The problem with much existing
software is that the physics and numerics are completely intertwined,
which makes it difficult to extend (and maintain). In Proteus the
description of the physical model and initial-boundary value problems
are nearly “method agnostic”. This approach has been used in the
developement of a variety of mathematical models and numerical
methods, both of which are described in more detail below
(Capabilities).

# Obtaining and Installing Proteus

For learning and experimenting there is a Docker image. Proteus can be installed through conda with::

```
% conda install proteus -c conda-forge
```

Proteus is available as source from our public GitHub repository. For a development installation, the installation of dependencies and the compilation of Proteus from source is done with::

```
% git clone https://github.com/erdc/proteus
% cd proteus
% conda env create -f environment-dev.yml
% conda activate proteus-dev
% pip install -v -e .
```

Alternatively, if you already have compilers (C,C++, and Fortran!) installed on your system, you can install Proteus through hashdist with the following commands::

```
% git clone https://github.com/erdc/proteus
% cd proteus
% make develop
% make test
```

More information is available on our Wiki, and you can ask for help on the Developers’ Mailing List.

# Running

If you have successfully compiled and tested Proteus then you should be able to do:

```
% cd $PROTEUS/tests/ci
% $PROTEUS_PREFIX/bin/parun poisson_3d_p.py poisson_3d_c0p1_n.py
```

The solution will be saved in a file ending in .xmf, which can be opened with ParaView or Ensight.

# Capabilities

Test problems and some analytical solutions have been implemented for

Poisson’s equation

The heat equation

Linear advection-diffusion-reaction equations

Singly degenerate nonlinear advection-diffusion-reaction equations (including various forms of Burger’s equation)

Doubly degenerate nonlinear advection-diffusion-reaction equations

The eikonal (signed distance) equation

The diffusive wave equations for overland flow

1D and 2D Shallow Water Equations

2D Dispersive Shallow Water Equations

Richards’ equation (mass conservative head- and saturation-based)

Two-phase flow in porous media with diffuse interface (fully coupled and IMPES formulations)

Two-phase flow in porous media with a sharp interface (level set formulation)

Stokes equations

Navier-Stokes equations

Reynolds-Averged Navier-Stokes equations

Two-phase Stokes/Navier-Stokes/RANS flow with a sharp interface (level set/VOF formulation)

Linear elasticity

These problems are solved on unstructured simplicial meshes. Simple meshes can be generated with tools included with Proteus, and more complex meshes can by imported from other mesh generators. The finite elements implemented are

Classical methods with various types of stabilization (entropy viscosity, variational multiscale, and algebraic methods)

\(C_0 P_1\)

\(C_0 P_2\)

\(C_0 Q_1\)

\(C_0 Q_2\)

Discontinuous Galerkin methods

\(C_{-1} P_0\)

\(C_{-1} P_1\) (Lagrange Basis)

\(C_{-1} P_2\) (Lagrange Basis)

\(C_{-1} P_k\) (Monomial Basis)

Non-conforming and mixed methods

\(P_1\) non-conforming

\(C_0 P_1 C_0 P_2\) Taylor-Hood

The time integration methods are

Backward Euler

Forward Euler

\(\Theta\) Methods

Strong Stability Preserving Runge-Kutta Methods

Adaptive BDF Methods

Pseudo-transient continuation

The linear solvers are

Jacobi

Gauss-Seidel

Alternating Schwarz

Full Multigrid

Wrappers to LAPACK, SuperLU, and PETSc

The nonlinear solvers are

Jacobi

Gauss-Seidel

Alternating Schwarz

Newton’s method

Nonlinear Multigrid (Full Approximation Scheme)

Fast Marching and Fast Sweeping

Additional tools are included for pre- and post-processings meshes and solutions files generated by Proteus and other models, including methods for obtaining locally-conservative velocity fields from \(C_0\) finite elements.

# Release Policy

The releases are numbered major.minor.revision

A revision increment indicates a bug fix or extension that shouldn’t break any models working with the same major.minor number.

A minor increment introduces significant new functionality but is mostly backward compatible

A major increment may require changes to input files and significantly change the underlying Proteus implementation.

These are not hard and fast rules, and there is no time table for releases.

# References

An Unstructured Finite Element Model for Incompressible Two-Phase Flow Based on a Monolithic Conservative Level Set Method (2020) M. Quezada de Luna, J. H. Collins, and C.E. Kees,

*International Journal for Numerical Methods in Fluids*.Robust explicit relaxation technique for solving the Green-Naghdi equations (2019) J.-L. Guermond, B. Popov, E. Tovar, C.E. Kees,

*Journal of Computational Physics*Preconditioners for Two-Phase Incompressible Navier-Stokes Flow (2019) N. Bootland, C.E. Kees, A. Wathen, A. Bentley

*SIAM Journal on Scientific Computing*, In Press.Modeling Sediment Transport in Three-Phase Surface Water Systems (2019) C.T. Miller, W.G. Gray, C.E. Kees, I.V. Rybak, B.J. Shepherd,

*Journal of Hydraulic Engineering*Fast Random Wave Generation in Numerical Tanks (2019) A. Dimakopoulos, T. de Lataillade, C.E. Kees,

*Proceedings of the Institution of Civil Engineers - Engineering and Computational Mechanics*, 1-29.A Partition of Unity Approach to Adaptivity and Limiting in Continuous Finite Element Methods (2019) D. Kuzmin, M. Quezada de Luna, C.E. Kees,

*Computers and Mathematics with Applications*.Simulating Oscillatory and Sliding Displacements of Caisson Breakwaters Using a Coupled Approach (2019) G. Cozzuto, A. Dimakopoulos, T. de Lataillade, P.O. Morillas, and C.E. Kees,

*Journal of Waterway, Port, Coastal, and Ocean Engineering*.A Monolithic Conservative Level Set Method with Built-In Redistancing (2019) M. Quezada de Luna, D. Kuzmin, C.E. Kees,

*Journal of Computational Physics*, 379, 262-278.Computational Model for Wave Attenuation by Flexible Vegetation (2018) S.A. Mattis, C.E. Kees, M.V. Wei, A. Dimakopoulos, and C.N. Dawson,

*Journal of Waterway, Port, Coastal, and Ocean Engineering*145(1), p.04018033.Well-Balanced Second-Order Finite Element Approximation of the Shallow Water Equations with Friction (2018) J.L. Guermond, M.Q. de Luna, B. Popov, C.E. Kees, and M.W. Farthing

*SIAM Journal on Scientific Computing*40(6), A3873-A3901.Dual-Scale Galerkin Methods for Darcy Flow (2018) G. Wang, G. Scovazzi, L. Nouveau, C.E. Kees, Simone Rossi, O. Colomes, and A. Main (2018)

*Journal of Computational Physics*354, 111-134.Implementation details of the level set two-phase Navier-Stokes equations in Proteus (2017) A. Bentley, N. Bootland, A. Wathen, C. Kees,

*Technical-Report-TR2017-10-ab.nb.aw.ck*.Evaluation of Galerkin and Petrov-Galerkin Model Reduction for Finite Element Approximations of the Shallow Water Equations (2017) A. Lozovsky, M. W. Farthing, and C.E. Kees,

*Computational Methods in Applied Mechanics and Engineering*318, 537-571.POD-Based Model Reduction for Stabilized Finite Element Approximations of Shallow Water Flows (2016) A. Lozovskiy, M.W. Farthing, C.E. Kees, E. Gildin

*Journal of Computational and Applied Mathematics*, 302, 50-70.An Immersed Structure Approach for Fluid-Vegetation Interaction (2015) S.A. Mattis, C.N. Dawson, C.E. Kees, M.W. Farthing,

*Advances in Water Resources*, 80,1-16.Finite Element Methods for Variable Density Flow And Solute Transport (2013) T.J. Povich, C.N. Dawson, M.W. Farthing, C.E. Kees

*Computational Geosciences*17(3), 529-549.Numerical simulation of water resources problems: Models, methods, and trends (2013) Cass T. Miller, Clint N. Dawson, Matthew W. Farthing, Thomas Y. Hou, Jingfang Huang, Christopher E. Kees, C.T. Kelley, and Hans Petter Langtangen

*Advances in Water Resources*, 51, 405-437,Numerical modeling of drag for flow through vegetated domains and porous structures (2012) S.A. Mattis, C. N. Dawson, C. E. Kees, and M. W. Farthing,

*Advances in Water Resources*, 39, pp44-59Parallel Computational Methods and Simulation for Coastal and Hydraulic Applications Using the Proteus Toolkit (2011) C. E. Kees and M. W. Farthing (2011)

*Supercomputing11: Proceedings of the PyHPC11 Workshop*A Conservative Level Set Method for Variable-Order Approximations and Unstructured Meshes (2011) C.E. Kees, I. Akkerman, Y. Bazilevs, and M. W. Farthing

*Journal of Computational Physics*230(12), pp4536–4558Locally Conservative, Stabilized Finite Element Methods For Variably Saturated Flow (2008) Kees, C.E., M. W. Farthing, and C. N. Dawson,

*Computer Methods in Applied Mechanics and Engineering*, 197, pp4610-4625Locally Conservative, Stabilized Finite Element Methods for a Class of Variable Coefficient Navier-Stokes Equations (2009) C. E. Kees, M. W. Farthing, and M. T. Fong,

*ERDC/CHL TR-09-12*Evaluating Finite Element Methods for the Level Set Equation (2009) M. W. Farthing and C. E. Kees,

*ERDC/CHL TR-09-11*A Review of Methods for Moving Boundary Problems (2009) C. E. Kees, M. W. Farthing, T. C. Lackey, and R. C. Berger,

*ERDC/CHL TR-09-10*Implementation of Discontinuous Galerkin Methods for the Level Set Equation on Unstructured Meshes (2008) M. W. Farthing and C. E. Kees,

*ERDC/CHL CHETN-XIII-2*