NumGy_LV02.qmd

---
title: "Numerical Simulation Methods in Geophysics, Part 2: Finite Differences"
subtitle: "1. MGPY+MGIN, 3. MDRS+MGEX-CMG"
author: "thomas.guenther@geophysik.tu-freiberg.de"
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# Some mathematical background
## Differential operators

* single derivative in space $\pdv{x}$ or time $\pdv t$

* gradient $\grad=(\pdv x, \pdv y, \pdv z)^T$

* divergence $\div \vb F = \pdv{F_x}{x} + \pdv{F_y}{y} + \pdv{F_z}{z}$

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Gauss': *what's in (volume) comes out (surface)*
$$\int_V \div\vb F\ dV = \iint_S \vb F \vdot \vb n\ dS$$
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![Gauss's theorem in EM](pics/Divergence_theorem_in_EM.svg)
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## Curl (rotation)

* curl $\curl \vb F = (\pdv{F_z}{y}-\pdv{F_y}{z}, \pdv{F_x}{z}-\pdv{F_z}{x}, \pdv{F_y}{x}-\pdv{F_x}{y})^T$

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Stoke: *what goes around comes around*
$$\int_S \curl \vb F \vdot \vb dS = \iint_S \vb F \vdot \vb dl$$
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![Stokes' theorem in EM](pics/Curl_theorem_in_EM.svg)
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:::{.fragment}
* curls have no divergence: $\div (\curl \vb F)=0$
:::

:::{.fragment}
* potential fields have no curl $\curl (\grad u)=0$ 
:::

# Numerical simulation
## Partial differential equations (PDEs)

Mostly: solution of PDEs for either scalar (potentials) or vectorial (fields) quantities

PDE Types ($u$-function, $f$-source, $a$/$c$-parameter):
 
* elliptic PDE: $\nabla^2 u=f$
* parabolic PDE $\nabla^2 u - a \frac{\partial u}{\partial t}=f$
* hyperbolic  $\nabla^2 u - c^2 \frac{\partial^2 u}{\partial t^2}=f$ (plus diffusive term)
$$\frac{\partial^2\ u}{{\partial x}^2} - c^2\frac{\partial^2 u}{\partial t^2} = 0$$
* coupled $\nabla\cdot u=f$ & $u = K \nabla p=0$ (Darcy flow)
* nonlinear $(\nabla u)^2=s^2$ (Eikonal equation)

## Poisson equation
potential field $u$ generates field $\vec{F}=-\nabla u$

causes some flow $\vec{j}=a \vec{F}$ 

$a$ is some sort of conductivity (electric, hydraulic, thermal)

continuity of flow: divergence of total current $\vb j + \vb j_s$ is zero

$$ \div (a \nabla u) = - \div \vb j_s $$

## Darcy's law

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volumetric flow rate $Q$ caused by gradient of pressure $p$

$$ Q = \frac{k A}{\mu L} \Delta p $$

$$\vb q = -\frac{k}{\mu} \nabla p $$

$$\div\vb q = -\div (k/\mu \grad p) = 0$$

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![Darcy's law](pics/darcy_law.svg)
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## The heat equation in 1D

sought: Temperature $T$ as a function of space and time

heat flux density $\vb q = \lambda\grad T$

$q$ in W/m², $\lambda$ - heat conductivity/diffusivity in W/(m.K)

Fourier's law: $\pdv{T}{t} - a \nabla^2 T = s$  ($s$ - heat source)

temperature conduction $a=\frac{\lambda}{\rho c}$ ($\rho$ - density, $c$ - heat capacity)

## Navier-Stokes equation

Stokes equation
$$ \mu \nabla^2 \vb v - \grad p + f = 0 $$

$$\div\vb v = 0 $$

Navier-Stokes equation
(incompressible, uniform viscosity)

$$ \pdv{\vb u}{t} +(\vb u \vdot \grad) \vb u = \nu \nabla^2 \vb u - 1/\rho \grad p + f $$

## Maxwell's equations

* Faraday's law: currents & varying electric fields $\Rightarrow$ magnetic field 
$$ \curl \vb H = \pdv{\vb D}{t} + \vb j $$ 
* Ampere's law: time-varying magnetic fields induce electric field
$$ \curl\vb E = -\pdv{\vb B}{t}$$ 
* $\div\vb D = \varrho$ (charge $\Rightarrow$), $\div\vb B = 0$ (no magnetic charge)
* material laws $\vb D = \epsilon \vb E$ and $\vec B = \mu \vb H$

## Helmholtz equations

$$ \nabla^2 \vb F + k^2 \vb F = f $$

results from wavenumber decomposition of diffusion or wave equations

approach: $\vb F = \vb{F_0}e^{\imath\omega t} \quad\Rightarrow\quad \pdv{\vb F}{t}=\imath\omega\vb F \quad\Rightarrow\quad \pdv[2]{\vb F}{t}=-\omega^2\vb F$

$$ \nabla^2 \vb F - a \nabla_t \vb F - c^2 \nabla^2_t \vb F = 0 $$

$$ \Rightarrow \nabla^2\vb F - a\imath\omega\vb F + c^2 \omega^2\vb F = 0 $$

## The Eikonal equation

Describes first-arrival times $t$ as a function of velocity ($v$) or slowness ($s$)

$$ |\grad t| = s = 1/v $$

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![](pics/tt0.png){.fragment width="458" height="420"}

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# The Finite Difference Method (FDM)
## Taylor expansion
Assume the Poisson equation $$\div(a\grad u)=f$$

Taylor expansion $$f(x)=f(x_0)+f'(x0)(x-x_0)+f''(x_0)(x-x_0)^2/2$$

## Finite differences

:::{.callout-tip icon="false" title="Approximate derivative operators by differences"} 
$$ \pdv{u}{x}\approx\frac{\Delta u}{\Delta x} $$
:::
and solution $u$ by finite values $u_i$ at points $x_i$, e.g.
$$ \dv*{u}{x}_{2.5} := (u_3-u_2) / (x_3-x_2) $$

<!-- $$ \dv*[2]{u}{x}_{3} :=   -->
$$\pdv[2]{u_3}{x}\approx 
\frac{\dv*{u}{x}_{3.5}-\dv*{u}{x}_{2.5}}{(x_4-x_2)/2} 
= \frac{(u_4-u_3)/(x_4-x_3)-(u_3-u_2)/(x_3-x_2)}{(x_4-x_2)/2} $$

## Difference stencil 
Assumption: equidistant discretization $\Delta x$, conductivity 1

1st derivative: $[-1, +1] / dx$, 2nd derivative $[+1, -2, +1] / dx^2$

Matrix-Vector product $\vb{A}\cdot\vb{u}=\vb{f}$ with

$$ \vb{A} = \begin{bmatrix}   
+1 & -2 & +1 & 0 & \ldots & &  \\
0 & +1 & -2 & +1 & 0 & \ldots & \\
\vdots & \vdots & \vdots & \ddots & \vdots &  \\
\ldots & \ldots & 0 & +1 & -2 & +1
\end{bmatrix}  $$

## FDM on the general Poisson equation
Assume the Poisson equation $$\div(a\grad u)=f$$

$$ \pdv{(a \dv*{u})}{z} = a\pdv[2]{u}{z} + \pdv{a}{z} \pdv{u}{z}  $$


## Boundary conditions

Dirichlet conditions: $u_0=u_B$ (homogeneous if 0)

Neumann conditions (homogeneous if 0)
$$ \pdv*{u}{x}_0=g_B $$

Mixed boundary conditions $u_0+\alpha du_0/dx=\gamma$

## Dirichlet BC implementation way 1
$u_0 = u_B$

$$ \begin{bmatrix}   
+1 & 0 & 0 & \ldots & &  \\
+1 & -2 & +1 & 0 & \ldots & \\
\vdots & \vdots & \ddots & \vdots &  \\
\ldots & \ldots & 0 & +1 & -2 & +1
\end{bmatrix} \cdot\vb{u} = 
\begin{bmatrix} u_B\\ f_1 \\ \vdots \\ f_N \end{bmatrix}  $$

## Dirichlet BC implementation way 2
$u_B - 2 u_1 + u_3 = f_1$

$$ \begin{bmatrix}   
-2 & +1 & 0 & \ldots & &  \\
+1 & -2 & +1 & 0 & \ldots & \\
\vdots & \vdots & \ddots & \vdots &  \\
\ldots & 0 & +1 & -2 & +1
\end{bmatrix} \cdot\vb{u} = 
\begin{bmatrix} f_1 - u_B\\ f_2 \\ \vdots \\ f_N \end{bmatrix} $$

## Neumann BC implementation way 1
$$ u_1 - u_0 = g_B$$ 

$$ \begin{bmatrix}   
-1 & +1 & 0 & \ldots & &  \\
+1 & -2 & +1 & 0 & \ldots & \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\ldots & \ldots & +1 & -2 & +1
\end{bmatrix} \cdot\vb{u} = 
\begin{bmatrix} f_0+g_B\\ f_1 \\ \vdots \\ f_N \end{bmatrix}  $$

## Neumann BC implementation way 2
$$u_0 - 2 u_1+u_2 = f_1 \qquad u_1 - u_0 = g_B \Rightarrow u_2-u_1=f_1+g_B $$ 
<!-- $-u_1 + u_2 = f_1 - g_B$ -->

$$ \begin{bmatrix}   
-1 & +1 & 0 & \ldots & &  \\
+1 & -2 & +1 & 0 & \ldots & \\
\vdots & \vdots & \ddots & \vdots &  \\
\ldots & 0 & +1 & -2 & +1
\end{bmatrix} \cdot\vb{u} = 
\begin{bmatrix} f_1 + g_B\\ f_2 \\ \vdots \\ f_N \end{bmatrix} $$

# Parabolic PDEs

## Heat transfer in 1D
$$ \pdv{T}{t} - a \pdv[2]{T}{z} = 0 $$

with the periodic boundary conditions:

* $T(z=0,t)=T_0 + \Delta T \sin \omega t$ (daily/yearly cycle)
* $\pdv{T}{z}(z=z_1) = 0$ (no change at depth)

and the initial condition $T(z, t=0)=\sin \pi z$ has the analytical solution
$$ T(z, t) = \Delta T e^{-\pi^2 t} \sin \pi z $$

## Explicit methods

$$ \pdv{T}{t} - a \pdv[2]{T}{z} = 0 $$

Solve Poisson equation $\div(a\grad u)=f$ 

for every time step $i$ (using FDM, FEM, FVM etc.)

Finite-difference step in time: update field by 
$$ T_{i+1} = T_i + a \pdv[2]{u}{z} \cdot \Delta t$$

## Implicit methods

next lecture

## Mixed methods