Welcome to the tutorial on solving the Helmholtz 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 http://www.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,
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 for a convergence plot and the visualisation of the results in Paraview or VisIt.
Installed and tested Nektar++ v5.6.0from 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-helmholtz.tar.gz
Unpack it using tar -xzvf basics-helmholtz.tar.gz to produce a directory
basics-helmholtz 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, Helm_mesh.geo;
a .msh file containing the mesh in Gmsh format, Helm_mesh.msh;
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 Helmholtz equation
| (1.1) |
where u is the independent variable. The Helmholtz equation can be solved in one, two and three spatial dimensions. We will here consider a two-dimensional problem.
The problem we want to solve consists of known boundary conditions and forcing function which depend on x and y. To model this problem we create a computational domain also referred to as mesh or grid (see section 2) on which we apply the following two-dimensional function with Dirichlet and Neumann boundary conditions.
| (1.2) |
where xb and yb represent the boundaries of the computational domain (see section 2.1) and λ is a positive constant.
We will set the boundary conditions and forcing function for this solver (see section 2.1) then, after running the solver (see section 3) we will post-process the data in order to visualise the results (see section 4).
![∇2u - λu = - (2π2 + λ) cos(πx )cos(πy ),
u(x,y) = cos(πx) cos(πy ),
u(xb = ±1, yb) = cos(πxb)cos(πyb),
∂ ∂
dnu(xb,yb = ±1) = ± dyu (xb,yb = ±1) = ∓ π[cos(πxb)sin(πyb)]](basics-helmholtz1x.svg)