update docs, add general, combined solver for heat diffusion equation (1D and 2D)

Author: LukasFuchs

README.md

@@ -10,31 +10,31 @@ The **Geod**ynamic **Mod**elling Tool**Box** is a Julia package primarily intend
 
 `GeoModBox.jl` includes a series of [exercises](./exercises/) and [examples](./examples/) of geodynamically well-defined problems. The exercises are provided as Jupyter notebooks for students to complete. The theoretical background is documented here.
 
-The solvers for each governing equation can be used separately or in combination for dimensional or non-dimensional problems, with only minimal modifications when calling the functions. Some typical initial conditions, such as a linearly increasing temperature, are predefined and can be called using [specific functions](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/Ini/).
+The solvers for each governing equation can be used separately or in combination for dimensional or non-dimensional problems, with only minimal modifications when calling the functions. For more informations on how to use the individual functions please see the [list of functions](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/listoffunctions/) or individual [exmples](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/Examples/). Some typical initial conditions, such as a linearly increasing temperature, are predefined and can be called using [specific functions](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/Ini/). In the following a brief explenation is given regarding the governing equations and the numerical method to solve them within the `GeoModBox.jl`. For more detailed information see the individual documentations. 
 
 ## Staggered Finite Difference
 
-To properly solve the governing equations, a staggered finite difference scheme is chosen for the *energy* and *momentum* equations. A staggered grid enables a correct and straightforward implementation of boundary conditions and ensures conservation of stress between nodes in cases of variable viscosity. This requires certain parameters to be defined on different grids.
+To properly solve the governing equations, a staggered finite difference scheme is chosen for the *energy* and *momentum* equations. A staggered grid enables a correct and straightforward implementation of boundary conditions and ensures conservation of stress between nodes in cases of variable viscosity. This requires certain parameters to be defined on different grids. For more information regarding the physical and numerical background, please refer to [this](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/GESolution/).
 
-Here, temperature, density, pressure, normal deviatoric stresses, and heat production rate are defined on the *centroids*. The deviatoric shear stresses are defined on the *vertices*, and velocities are defined between the *vertices*. Viscosity is required on both.
+Within the `GeoModBox.jl`, temperature, density, pressure, normal deviatoric stresses, and heat production rate are defined on the *centroids*. The deviatoric shear stresses are defined on the *vertices*, and velocities are defined between the *vertices*. Viscosity is required on both.
 
-For further details on the implementation in `GeoModBox.jl`, see [here](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/GESolution/).
+For further details on the implementation in `GeoModBox.jl`, see the individual documentations for each governing equation. 
 
 ## Energy Conservation Equation
 
-In geodynamics, the energy is described by the temperature and needs to be conserved within a closed system. Here, we solve the *temperature conservation equation*, or *temperature equation*, using an *operator splitting* method, that is, we first solve the *advective* part of the temperature equation, followed by the *diffusive* part. 
+In geodynamics, the energy is described by the temperature and needs to be conserved within a closed system. Within the `GeoModBox.jl`, the *temperature conservation equation*, or *temperature equation*, is solved using an *operator splitting* method, that is, first the *advective* part of the temperature equation is solved, followed by the *diffusive* part. 
 
 ### [Heat Diffusion Equation](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/DiffMain/)
 
-`GeoModBox.jl` provides several finite difference schemes for solving the *diffusive part* of the time-dependent or steady-state temperature equation, including radioactive heating, in both [1-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/DiffOneD/) and [2-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/DiffTwoD/). The solvers are located in [src/HeatEquation](./src/HeatEquation/). Currently, only *Dirichlet* and *Neumann* thermal boundary conditions are supported. Most functions assume constant thermal parameters (with the exception of the 1-D solvers and the 2-D defect correction solver).
+`GeoModBox.jl` provides several finite difference schemes for solving the *diffusive part* of the time-dependent or steady-state temperature equation, including radioactive heating, in both [1-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/DiffOneD/) and [2-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/DiffTwoD/). The solvers are located in [src/HeatEquation](./src/HeatEquation/). Currently, only *Dirichlet* and *Neumann* thermal boundary conditions are supported. Most functions assume constant thermal parameters (with the exception of the 1-D solvers and the 2-D, iterative implicit solver, called **iterative defection correction method**).
 
 ### [Heat Advection Equation](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/AdvectMain/)
 
-```GeoModBox.jl``` provides various methods to advect properties within the model domain. The routines are structured so that any property defined on *centroids* (including *ghost nodes* at all boundaries) can be advected using the described solvers. Using passive tracers, one may choose to advect either the absolute temperature or the phase ID.
+`GeoModBox.jl` provides various methods to advect properties within the model domain. The routines are structured so that any property defined on *centroids* (including *ghost nodes* at all boundaries) can be advected using the described solvers. Using passive tracers, one may choose to advect either the absolute temperature or the phase ID.
 
 ## [Momentum Conservation Equation](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/MomentumMain/)
 
-On geological timescales, Earth's mantle and lithosphere deform slowly due to their high viscosity, allowing us to neglect inertial forces. This simplifies the Navier-Stokes equation into the **Stokes equation**. `GeoModBox.jl` provides two main methods to solve the Stokes equation in [1-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/MomentumOneD/) and [2-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/MomentumTwoD/): the direct method and the defect correction method, applicable for both constant and variable viscosity fields. Velocity and pressure are defined on a staggered grid, and ghost nodes are included to ensure proper implementation of free-slip and no-slip boundary conditions. 
+On geological timescales, Earth's mantle and lithosphere deform slowly due to their high viscosity, allowing us to neglect inertial forces. This simplifies the Navier-Stokes equation into the **Stokes equation**. `GeoModBox.jl` provides two main methods to solve the Stokes equation in [1-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/MomentumOneD/) and [2-D](https://geosci-ffm.github.io/GeoModBox.jl/dev/man/MomentumTwoD/): the direct method and the defection correction method, applicable for both constant and variable viscosity fields. Velocity and pressure are defined on a staggered grid, and ghost nodes are included to ensure proper implementation of free-slip and no-slip boundary conditions.
 
 ## [Benchmarks and Examples](./examples/)
 
@@ -44,7 +44,7 @@ The following are visualizations of selected examples provided by `GeoModBox.jl`
 
 <img src="./examples/DiffusionEquation/2D/Results/Gaussian_Diffusion_CNA_nx_100_ny_100.gif" alt="drawing" width="600"/>
 
-**Figure 1. Gaussian Diffusion.** Time-dependent, diffusive solution of a 2-D Gaussian temperature anomaly at a resolution of 100 × 100, using the [Crank-Nicholson approach](./src/HeatEquation/2Dsolvers.jl), compared to the analytical solution.  
+**Figure 1. Gaussian Diffusion.** Time-dependent, diffusive solution of a 2-D Gaussian temperature anomaly at a resolution of 100 × 100, using the [Crank-Nicholson approach](./src/HeatEquation/2Dsolvers.jl),  compared to the analytical solution.  
 Top Left: 2-D temperature field with numerical isotherms (solid black) and analytical isotherms (dashed yellow).  
 Top Right: Total deviation from the analytical solution.  
 Bottom Left: 1-D y-profile along $x = 0$.  

docs/make.jl

@@ -16,12 +16,12 @@ makedocs(
         "Governing Equation" => Any[ # Checked! 
             "Solution" => Any[
                 "General" => "man/GESolution.md", # checked!
-                "Initial Condition" => "man/Ini.md", # Checked!
+                "Specifications" => "man/Ini.md", # Checked!
             ],
             "Heat Diffusion Equation" => Any[
-                "General" => "man/DiffMain.md", # checked!
-                "1D" => "man/DiffOneD.md", # checked!
-                "2D" => "man/DiffTwoD.md", # checked! 
+                "General" => "man/DiffMain.md", # checked! Checked!! Checked!!!
+                "1D" => "man/DiffOneD.md", # checked! Checked!! Checked!!!
+                "2D" => "man/DiffTwoD.md", # checked! Checked!!!
             ],
             "Advection Equation" => Any[
                 "General" => "man/AdvectMain.md", # Checked! 

docs/src/assets/Gaussian_ResTest.png

@@ -8,31 +8,31 @@ The **Geod**ynamic **Mod**elling Tool**Box** is a Julia package primarily intend
 
 `GeoModBox.jl` includes a series of [exercises](https://github.com/GeoSci-FFM/GeoModBox.jl/blob/main/exercises/) and [examples](./man/Examples.md) of geodynamically well-defined problems. The exercises are provided as Jupyter notebooks for students to complete. The theoretical background is documented here.
 
-The solvers for each governing equation can be used separately or in combination for dimensional or non-dimensional problems, with only minimal modifications when calling the functions. Some typical initial conditions, such as a linearly increasing temperature, are predefined and can be called using [specific functions](./man/Ini.md).
+The solvers for each governing equation can be used separately or in combination for dimensional or non-dimensional problems, with only minimal modifications when calling the functions. For more informations on how to use the individual functions please see the [list of functions](./man/listoffunctions.md) or individual [exmples](./man/Examples.md). Some typical initial conditions, such as a linearly increasing temperature, are predefined and can be called using [specific functions](./man/Ini.md). In the following a brief explenation is given regarding the governing equations and the numerical method to solve them within the `GeoModBox.jl`. For more detailed information see the individual documentations. 
 
 ## Staggered Finite Difference
 
-To properly solve the governing equations, a staggered finite difference scheme is chosen for the *energy* and *momentum* equations. A staggered grid enables a correct and straightforward implementation of boundary conditions and ensures conservation of stress between nodes in cases of variable viscosity. This requires certain parameters to be defined on different grids.
+To properly solve the governing equations, a staggered finite difference scheme is chosen for the *energy* and *momentum* equations. A staggered grid enables a correct and straightforward implementation of boundary conditions and ensures conservation of stress between nodes in cases of variable viscosity. This requires certain parameters to be defined on different grids. For more information regarding the physical and numerical background, please refer to [this](./man/GESolution.md).
 
-Here, temperature, density, pressure, normal deviatoric stresses, and heat production rate are defined on the *centroids*. The deviatoric shear stresses are defined on the *vertices*, and velocities are defined between the *vertices*. Viscosity is required on both.
+Within the `GeoModBox.jl`, temperature, density, pressure, normal deviatoric stresses, and heat production rate are defined on the *centroids*. The deviatoric shear stresses are defined on the *vertices*, and velocities are defined between the *vertices*. Viscosity is required on both.
 
-For further details on the implementation in `GeoModBox.jl`, see [here](./man/GESolution.md).
+For further details on the implementation in `GeoModBox.jl`, see the individual documentations for each governing equation. 
 
 ## Energy Conservation Equation
 
-In geodynamics, the energy is described by the temperature and needs to be conserved within a closed system. Here, we solve the *temperature conservation equation*, or *temperature equation*, using an *operator splitting* method, that is, we first solve the *advective* part of the temperature equation, followed by the *diffusive* part. 
+In geodynamics, the energy is described by the temperature and needs to be conserved within a closed system. Within the `GeoModBox.jl`, the *temperature conservation equation*, or *temperature equation*, is solved using an *operator splitting* method, that is, first the *advective* part of the temperature equation is solved, followed by the *diffusive* part. 
 
 ### [Heat Diffusion Equation](./man/DiffMain.md)
 
-`GeoModBox.jl` provides several finite difference schemes for solving the *diffusive part* of the time-dependent or steady-state temperature equation, including radioactive heating, in both [1-D](./man/DiffOneD.md) and [2-D](./man/DiffTwoD.md). The solvers are located in [src/HeatEquation](https://github.com/GeoSci-FFM/GeoModBox.jl/blob/main/src/HeatEquation/). Currently, only *Dirichlet* and *Neumann* thermal boundary conditions are supported. Most functions assume constant thermal parameters (with the exception of the 1-D solvers and the 2-D defect correction solver).
+`GeoModBox.jl` provides several finite difference schemes for solving the *diffusive part* of the time-dependent or steady-state temperature equation, including radioactive heating, in both [1-D](./man/DiffOneD.md) and [2-D](./man/DiffTwoD.md). The solvers are located in [src/HeatEquation](https://github.com/GeoSci-FFM/GeoModBox.jl/blob/main/src/HeatEquation/). Currently, only *Dirichlet* and *Neumann* thermal boundary conditions are supported. Most functions assume constant thermal parameters (with the exception of the 1-D solvers and the 2-D, iterative implicit solver, called **iterative defection correction method**).
 
 ### [Heat Advection Equation](./man/AdvectMain.md)
 
-```GeoModBox.jl``` provides various methods to advect properties within the model domain. The routines are structured so that any property defined on *centroids* (including *ghost nodes* at all boundaries) can be advected using the described solvers. Using passive tracers, one may choose to advect either the absolute temperature or the phase ID.
+`GeoModBox.jl` provides various methods to advect properties within the model domain. The routines are structured so that any property defined on *centroids* (including *ghost nodes* at all boundaries) can be advected using the described solvers. Using passive tracers, one may choose to advect either the absolute temperature or the phase ID.
 
 ## [Momentum Conservation Equation](./man/MomentumMain.md)
 
-On geological timescales, Earth's mantle and lithosphere deform slowly due to their high viscosity, allowing us to neglect inertial forces. This simplifies the Navier-Stokes equation into the **Stokes equation**. `GeoModBox.jl` provides two main methods to solve the Stokes equation in [1-D](./man/MomentumOneD.md) and [2-D](./man/MomentumTwoD.md): the direct method and the defect correction method, applicable for both constant and variable viscosity fields. Velocity and pressure are defined on a staggered grid, and ghost nodes are included to ensure proper implementation of free-slip and no-slip boundary conditions. 
+On geological timescales, Earth's mantle and lithosphere deform slowly due to their high viscosity, allowing us to neglect inertial forces. This simplifies the Navier-Stokes equation into the **Stokes equation**. `GeoModBox.jl` provides two main methods to solve the Stokes equation in [1-D](./man/MomentumOneD.md) and [2-D](./man/MomentumTwoD.md): the direct method and the defection correction method, applicable for both constant and variable viscosity fields. Velocity and pressure are defined on a staggered grid, and ghost nodes are included to ensure proper implementation of free-slip and no-slip boundary conditions. 
 
 ## [Benchmarks and Examples](./man/Examples.md)
 

docs/src/index.md

@@ -34,7 +34,7 @@ $\begin{equation}
 
 which represents the number of grid points traversed in a single time step. 
 
-Unfortunately, this scheme is unconditionally unstable for the advection equation, as shown by a *Von Neumann* or *Hirt's stability analysis*. Furthermore, the central difference at $i$ causes amplification of the variable (here, temperature) at each subsequent time step. Hence, the solution continually grows and is unstable.
+Unfortunately, this scheme is unconditionally unstable for the advection equation, as shown by a *Von Neumann* or *Hirt's stability analysis*. The central difference at $i$ causes amplification of the variable (here, temperature) at each subsequent time step. Hence, the solution continually grows and is unstable.
 
 ## Lax-Friedrichs method
 
@@ -99,7 +99,7 @@ where $x_i$ is the coordinate of the Eulerian grid point $i$, $\Delta{t}$ is the
 
 Interpolate the temperature at $t_n$ from the surrounding Eulerian grid points onto the position $X_i$, e.g., using `cubic_spline_interpolation()`.
 
-**3. Update the Temperature field**
+**3. Update the temperature field**
 
 Assuming the temperature at the grid point at $t_{n+1}$ equals the interpolated value at $X_i$ at time $t_n$: