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NumGy_LV07E.qmd
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---
title: "Numerical Simulation Methods in Geophysics, Exercise 7:<br> Timestepping with FE"
subtitle: "1. MGPY+MGIN"
author: "[email protected]"
title-slide-attributes:
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include-in-header:
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<script>
window.MathJax = {
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load: ['[tex]/physics']
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# jupyter: python3
---
# Recap Poisson and heat equations
## Recap
* [x] solve the Poisson equation for arbitrary $x$ and $a$
* [x] sources and $a$ contrasts cause curvature in $u$<br>
* positive source or $a$ increase $\Rightarrow$ negative u" $\Rightarrow$ maximum
* single $f$ $\Rightarrow$ piecewise linear, full $f$ $\Rightarrow$ parabola
* [x] Dirichlet BC determine shift (& slope if double)
* [x] Neumann BC determine slope of $u$
* [x] accuracy (compare analytical) depends on discretization
* [x] now go for instationary (parabolic) problem by time stepping
* curvature in $u$ causes negative change of $u$
# Time stepping
## Time stepping - explicit method
$$ \pdv{T}{t} - a \pdv[2]{T}{z} = 0 $$
Finite-difference approximation
$$ \pdv{T}{t}^n \approx \frac{T^{n+1}-T^n}{\Delta t} = a \pdv[2]{T^n}{z} $$
## Explicit
:::: {.columns}
::: {.column}
Start $T^0$ with initial condition (e.g. 0)
Update field by
$$ T^{n+1} = T^n + a \pdv[2]{T^n}{z} \cdot \Delta t$$
E.g. by using the matrix $A$
:::
::: {.column}
:::
::::
## Implicit methods
:::: {.columns}
::: {.column}
$$ \pdv{T}{t}^{n+1} \approx \frac{T^{n+1}-T^n}{\Delta t} = a \pdv[2]{T}{z} ^{n+1} $$
$$ \frac{1}{\Delta t} T^{n+1} - a \pdv[2]{T}{z} ^{n+1} = \frac{1}{\Delta t} T^n $$
$$ (\vb M - \vb A) \vb u^{n+1} = \vb M \vb u^n $$
$\vb M$ - mass matrix
:::
::: {.column}
:::
::::
## Mixed - Crank-Nicholson method
$$ \pdv{T}{t}^{n+1/2} \approx \frac{T^{n+1}-T^n}{\Delta t} = \frac12 a \pdv[2]{T}{z} ^n + \frac12 a \pdv[2]{T}{z} ^{n+1} $$
:::: {.columns}
::: {.column width="60%"}
$$ \frac{2}{\Delta t} T^{n+1} - a \pdv[2]{T^{n+1}}{z} =
\frac{2}{\Delta t} T^n + a \pdv[2]{T^n}{z} $$
$$ (2\vb M - \vb A) \vb u^{n+1} = (2\vb M + \vb A) \vb u^n $$
<!-- $$ \pdv{T}{t}^n \approx \frac{T^{n+1}-T^n}{\Delta t} = \frac12 a \pdv[2]{T}{z} ^n + \frac12 a \pdv[2]{T}{z} ^{n+1} $$ -->
:::
::: {.column width="40%"}
:::
::::
# Report
## Task
1. Complete functions delivering both stiffness matrix and right-hand-side vector using FE discretizations
1. Use a non-equidistant discretization of the Earth with increasing layer thicknesses (choose and substantiate).
1. Solve instationary heat equation with periodic boundary condition (yearly cycle) for the Earth using a constant but meaningfull thermal diffusivity.
1. Compare the solutions using explicit, implicit and mixed timestepping methods with the analytical solution.
## Questions
* Interpret the results in terms of physical behaviour. How does a change in the diffusivity affect the result.
* Is there a difference between FD and FE discretizations? Why (not)?
* Make a statement about the stability and accuracy of the methods.
* After which time approaches the numeric solution the analytical one?
* How can you evaluate the numerical accuracy if there is not analytical solution?
## Deliverables
Format can be Jupyter Notebook and/or PDF
Complete codes to run the results