Introduction

Welcome to the tutorial of the Advection problem using the Advection-Diffusion-Reaction (ADR) Solver in the Nektar++ framework. This tutorial is aimed to show the main features of the ADR solver in a simple manner. If you have not already downloaded and installed Nektar++, please do so by visiting nektar.info, where you can also find the User-Guide with the instructions to install the library.

This tutorial requires:

- Nektar++ ADRSolver and pre- and post-processing tools,
- the open-source mesh generator Gmsh,
- the visualisation tool Paraview or VisIt

After the completion of this tutorial, you will be familiar with:

- the generation of a simple mesh in Gmsh and its conversion into a Nektar++-compatible format;
- the visualisation of the mesh in Paraview or VisIt
- the setup of the initial and boundary conditions, the parameters and the solver settings;
- running a simulation with the ADR solver; and
- the post-processing of the data and the visualisation of the results in Paraview or VisIt.

- Installed and tested
*Nektar++*v5.3.0from a binary package, or compiled it from source. By default binary packages will install all executables in`/usr/bin`

. If you compile from source they will be in the sub-directory`dist/bin`

of the`build`

directory you created in the*Nektar++*source tree. We will refer to the directory containing the executables as`$NEK`

for the remainder of the tutorial. - Downloaded the tutorial files: basics-advection-diffusion.tar.gz

Unpack it using`tar -xzvf basics-advection-diffusion.tar.gz`

to produce a directory`basics-advection-diffusion`

with subdirectories called`tutorial`

and`complete`

.We will refer to the

`tutorial`

directory as`$NEKTUTORIAL`

.The tutorial folder contains:

- a Gmsh file to generate the mesh,
`ADR_mesh.geo`

; - a .msh file containing the mesh in Gmsh format,
`ADR_mesh.msh`

;

- a Gmsh file to generate the mesh,

The ADR solver can solve various problems, including the unsteady advection, unsteady diffusion, unsteady advection diffusion equation, etc. For a more detailed description of this solver, please refer to the User-Guide.

In this tutorial we focus on the unsteady advection equation

| (1.1) |

where u is the independent variable and V = [V_{x} V_{y} V_{z}] is the advection velocity. The
unsteady advection equation can be solved in one, two and three spatial dimensions. We will
here consider a two-dimensional problem, so that V = [V_{x} V_{y}].

The problem we want to run consists of a given initial condition (which depends on x and y) travelling in the x-direction at a constant advection velocity. To model this problem we create a computational domain also referred to as mesh or grid (see section 2.1) on which we apply the following two-dimensional function as initial condition and periodic as well as time-dependent Dirichlet boundary conditions at the mesh boundaries

| (1.2) |

where x_{b} and y_{b} represent the boundaries of the computational domain (see section 2.2),
V_{x} = 2,V_{y} = 0 and κ = 2π.

We successively setup the other parameters of the problem, such as the time-step, the time-integration scheme, the I/O configuration, etc. (see section 2.2). We finally run the solver (see section 3) and post-process the data in order to visualise the results (see section 4).