Chapter 1
Introduction

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:

Goals

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

Task: 1.1 Prepare for the tutorial. Make sure that you have:

Task: 1.2 Additionally, you should also install

1.1 Background

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

∇2u - λu =  f,
(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.

1.2 Problem description

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.

 2              2
∇ u - λu = - (2π + λ) cos(πx )cos(πy ),

u(x,y) = cos(πx) cos(πy ),

u(xb = 1, yb) = cos(πxb)cos(πyb),

∂-u(xb,yb = 1) =  ∂-u (xb,yb = 1) = ∓ π[cos(πxb)sin(πyb)]
dn                  dy
(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).