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Adaptive Finite Difference Methods and Implementation in the Software DUNE.

Ediza Dabatos, Krane Kint LU (2024) In Bachelor’s Theses in Mathematical Sciences NUMK11 20241
Centre for Mathematical Sciences
Mathematics (Faculty of Sciences)
Abstract
The diffusion equation is a partial differential equation that describes how
quantities such as heat, particles, or other diffusive substances spread out over
time and space. This type of equation can be, at often times, quite challenging
to solve analytically. Finding the exact solution is often difficult due to the com-
plexity of the functions involved. This thesis focuses on solving the diffusion
equation using an adaptive finite difference method. We will be exploring two
different methods that would produce a numerical solution. The concept of ad-
aptivity optimizes our solution in a way that enhances accuracy and decreases
both computational power and time. We will also be working with the DUNE
software which is... (More)
The diffusion equation is a partial differential equation that describes how
quantities such as heat, particles, or other diffusive substances spread out over
time and space. This type of equation can be, at often times, quite challenging
to solve analytically. Finding the exact solution is often difficult due to the com-
plexity of the functions involved. This thesis focuses on solving the diffusion
equation using an adaptive finite difference method. We will be exploring two
different methods that would produce a numerical solution. The concept of ad-
aptivity optimizes our solution in a way that enhances accuracy and decreases
both computational power and time. We will also be working with the DUNE
software which is particularly helpful in creating adaptive grids. The software
enables us to access the grid cells and it’s neighbors which will be useful when
creating our solver. (Less)
Popular Abstract
As an artist that paints either portraits or landscapes, it is important to choose the
right brush. If we want, for example, to paint a blue sky, then one would in general
choose a wide brush with big coverage. Details, however, require intricate and precise
work. An artist would then prefer smaller and thinner brushes.
This idea of efficiently choosing the right tool, is widely used to handle complex math-
ematical problems. In particular, a partial differential equation that is capable of de-
scribing the physical world can be proven challenging to solve. This is because solv-
ing such equations analytically, involves an infinite number of points in a continuous
space. Meaning that although exact solutions exist, they are often... (More)
As an artist that paints either portraits or landscapes, it is important to choose the
right brush. If we want, for example, to paint a blue sky, then one would in general
choose a wide brush with big coverage. Details, however, require intricate and precise
work. An artist would then prefer smaller and thinner brushes.
This idea of efficiently choosing the right tool, is widely used to handle complex math-
ematical problems. In particular, a partial differential equation that is capable of de-
scribing the physical world can be proven challenging to solve. This is because solv-
ing such equations analytically, involves an infinite number of points in a continuous
space. Meaning that although exact solutions exist, they are often exhausting and even,
at worst cases, nearly impossible to get.
To make this process simpler and more practical, we will use a method called the finite
difference method, which breaks down the continuous space into a grid of points to
create an approximated solution with reasonable accuracy. This approach gives us a
mean to attack the problem in a more direct and manageable way. Furthermore, by
introducing an adaptive ideology of carefully choosing the appropriate points to solve
the problem on, we can be more efficient to create such approximated solution.
Depending on the number of grid points, we can, in general, increase the accuracy
of our solution. It is however important to note that increasing the data points re-
quires more computational work and time. So increasing the number of grid points
irresponsibly, defeats the purpose of a simple and practical work. Although we would
get a finer and detailed painting using only a thin and small brush, it would take an
unreasonable amount of time before we can finish a painting.
This implies that we need to be strategic on how and where to increase the number of
grid points. One particular strategy is to carefully study the problem, with the hope of
identifying the areas such that the values do not change drastically. This is analogous
to the sky painting where we would only use a wide brush. Our painting canvas in
this case would be our domain and our solution is the painting. We would want our
solver, the program that we will create to solve the diffusion problem, to know which
areas it should use a wide brush and which areas a small brush is needed. (Less)
Please use this url to cite or link to this publication:
author
Ediza Dabatos, Krane Kint LU
supervisor
organization
alternative title
Adaptiva Finita Differensmetoder och Implementeringen i DUNE.
course
NUMK11 20241
year
type
M2 - Bachelor Degree
subject
keywords
PDE, Partial Differential Equations, FDM, Finite Difference Methods, Adaptive Finite Difference Methods, DUNE, Diffusion Equation, Poisson Equation, Vibrating System
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFNA-4059-2024
ISSN
1654-6229
other publication id
2024:K25
language
English
id
9178881
date added to LUP
2025-06-12 15:06:32
date last changed
2025-06-12 15:06:32
@misc{9178881,
  abstract     = {{The diffusion equation is a partial differential equation that describes how
quantities such as heat, particles, or other diffusive substances spread out over
time and space. This type of equation can be, at often times, quite challenging
to solve analytically. Finding the exact solution is often difficult due to the com-
plexity of the functions involved. This thesis focuses on solving the diffusion
equation using an adaptive finite difference method. We will be exploring two
different methods that would produce a numerical solution. The concept of ad-
aptivity optimizes our solution in a way that enhances accuracy and decreases
both computational power and time. We will also be working with the DUNE
software which is particularly helpful in creating adaptive grids. The software
enables us to access the grid cells and it’s neighbors which will be useful when
creating our solver.}},
  author       = {{Ediza Dabatos, Krane Kint}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor’s Theses in Mathematical Sciences}},
  title        = {{Adaptive Finite Difference Methods and Implementation in the Software DUNE.}},
  year         = {{2024}},
}