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Copy file name to clipboardExpand all lines: README.md
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@@ -7,7 +7,7 @@ The **Geod**ynamic **Mod**elling Tool**Box** is a julia package mainly used for
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2)[**momentum**](./docs/src/man/MomentumMain.md),
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3)[**mass** and **compositon**](./docs/src/man/AdvectMain.md).
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The ```GeoModBox.jl``` includes a series of [exercises](/exercises/) and [examples](/examples/) of different geodynamically well defined problems. The exercises are given as Jupyter notebooks for the students to complete. The theoretical background and detailed explenations of the examples and functions are mainly given in the [documentation](https://lukasfuchs.github.io/GeoModBox.jl/).
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The ```GeoModBox.jl``` includes a series of [exercises](/exercises/) and [examples](/examples/) of different geodynamically well defined problems. The exercises are given as Jupyter notebooks for the students to complete. The solutions of the exercises are available upon request. The goal of the course is to teach the students the advantages and disadvantages of certain finite difference scheme, to combine different solution techniques and to finally build a two-dimensional, thermal convection model. The theoretical background and detailed explenations of the examples and functions are mainly given in the [documentation](https://geosci-ffm.github.io/GeoModBox.jl/).
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- Different properties can be advected!
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- Defect correction
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## Structure
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<!--## Structure
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<!--The ```GeoModBox.jl``` is -->
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The ```GeoModBox.jl``` is a two-dimensional, staggered, finite difference code to solve the governing equations for certain geodynamical problems (so far, only for linear viscous problems). One can use the solvers for the *temperature*, *momentum*, and *mass* equations seperately or couple them using a so-called *operator splitting* method.
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<!-- ## Initial Conditions -->
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In any case, first, one needs to setup the specific model parameters, that is the geometry (xmin,xmax,ymin,ymax), the grid (e.g., nc, nv, $\Delta$), and the physical constants (e.g., $\rho$, $\eta$).
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<!---
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- Set Up, Geometry, Grid, etc.
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- IniTemperature, IniVelocity, IniPhase,
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- Tracer Initialization
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- Solution of Stokes equation, Diffusion equation, Advection Equation,
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- ...
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-->
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Second, one needs to allocate the array of each needed variable. Third, one needs to define the boundary conditions and the initial time stepping. Before the time loop, one also needs to initialize the tracers and the parameters for the system of equations. Some initial conditions (temperature, density, velocity) can be setup, either via the marker funktion ```IniTracer2D()``` or the functions located in 2Dini.jl.
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<!-- ## Scaling -->
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Within the time loop, one first solves the *momentum* equation, followed by the reevaluation of the time step. Second, one solves the *advection* part of the *temperature equation*, followed by its *diffusion* part. All figures can either be stored for certain time steps or as *gif* animations of the entire problem.
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For more details on how to use the different functions, please see the [examples](./examples/), [exercises](./exercises/), and [documentation](https://geosci-ffm.github.io/GeoModBox.jl/).
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<!-- ## Scaling -->
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## [Benchmarks and Examples](./examples/)
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In the following, some highlights of the ```GeoModBox.jl``` are shown. For more details on the examples and benchmarks, please see the [documentation](https://geosci-ffm.github.io/GeoModBox.jl/).
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### [Gaussian Temperature Diffusion](./examples/DiffusionEquation/2D/Gaussian_Diffusion.jl)
**Figure 1. Gaussian Diffusion.** Time-dependent, diffusive solution of a 2-D Gaussian temperature anomaly using the [Crank-Nicholson approach](./src/HeatEquation/CNA.jl) in comparison to its analytical solution. Top Left: 2-D temperature field of the numerical solution and isotherms lines of the numerical (solid black) and analytical (dashed yellow) solution. Top Right: Total deviation to the analytical solution. Bottom Left: 1-D y-profile along x=0. Bottom Right: Root Mean Square total devation of the temperature over time.
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**Figure 1. Gaussian Diffusion.** Time-dependent, diffusive solution of a 2-D Gaussian temperature anomaly using the [Crank-Nicholson approach](./src/HeatEquation/2Dsolvers.jl) in comparison to its analytical solution. Top Left: 2-D temperature field of the numerical solution and isotherms lines of the numerical (solid black) and analytical (dashed yellow) solution. Top Right: Total deviation to the analytical solution. Bottom Left: 1-D y-profile along x=0. Bottom Right: Root Mean Square total devation of the temperature over time.
**Figure 3. Rigid-Body-Rotation.** Time-dependent solution of a rotating circular temperature anomaly using the **upwind (first)**, **semi-lagrangian (second)**, and **tracer (third)** method. Within a circular area of our model domain the velocity is set to the velocity of a rigid rotation and outside euqal to zero. The temperature is scaled by the maximum temperature of the anomaly.
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**Figure 3. Rigid-Body-Rotation.** Time-dependent solution of a rotating circular temperature anomaly using the **upwind (first)**, **semi-lagrangian (second)**, and **tracer (third)** method. Within a circular area of our model domain the velocity is set to the velocity of a rigid rotation and outside euqal to zero. The temperature is scaled by the maximum temperature of the anomaly. Empty cells are not refilled with markers.
Copy file name to clipboardExpand all lines: docs/src/index.md
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# GeoModBox.jl
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The **Geod**ynamic **Mod**elling Tool**Box** is a julia package mainly used for teaching purposes. The package provides different finite difference, staggered, discretization schemes to numerically solve the governing equations for a two-dimensional geodynamic problem. The governing equations are the conservation equations of
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1)[**energy**](./man/DiffMain.md),
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2)[**momentum**](./man/MomentumMain.md),
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3)[**mass** and **compositon**](./man/AdvectMain.md).
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The ```GeoModBox.jl``` includes a series of [exercises](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/) and [examples](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/) of different geodynamically well defined problems. The exercises are given as Jupyter notebooks for the students to complete. The theoretical background is mainly given here in the documentation.
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------------------
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## Staggered Finite Difference
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The ```GeoModBox.jl``` includes a series of [exercises](https://github.com/GeoSci-FFM/GeoModBox.jl/blob/main/exercises/) and [examples](https://github.com/GeoSci-FFM/GeoModBox.jl/blob/main/examples/) of different geodynamically well defined problems. The exercises are given as Jupyter notebooks for the students to complete. The theoretical background is mainly given here in the documentation.
To properly solve the governing equations, a staggered finite difference scheme is choosen for the *energy* and *momentum* equations. A staggered grid enables a correct, straight forward implementation of certain boundary conditions and enables the conservation of stress between the nodes in case of a variable viscosity. This also requires that certain parameters are defined on different grids.
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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.
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Here, the temperature, density, pressure are defined on the *centroids*, ... are defined on the *vertices*, the velocities are defined in between the *vertices*, and the viscosity is needed on both.
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------------------
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For more details on how this is used in the ```GeoModBox.jl``` see [here](./man/GESolution.md).
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### [Heat Diffusion Equation](./man/DiffOneD.md)
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##Energy Conservation Equation
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The ```GeoModBox.jl``` provides different finite difference (**FD**) schemes (e.g., [forward](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/HeatEquation/ForwardEuler.jl)and [backward](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/HeatEquation/BackwardEuler.jl) Euler, [Crank-Nicholson approach](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/HeatEquation/CNA.jl), [ADI](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/HeatEquation/ADI.jl)) to solve the *diffusive part* of the time-dependent or steady-state *temperature equation* including radioactive heating, in [1-D](./man/DiffOneD.md) and [2-D](./man/DiffTwoD.md). The solvers are located in [src/HeatEquation](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/HeatEquation/). So far, only *Dirichlet* and *Neumann* thermal boundary conditions are available. Currently, most of the functions assume constant thermal parameters (except for the 1-D solvers).
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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.
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The examples of solving the *heat diffusion equation* include, amongst others:
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- the determination of an [oceanic](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/DiffusionEquation/1D/OceanicGeotherm_1D.jl) and [continental](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/DiffusionEquation/1D/ContinentalGeotherm_1D.jl) 1-D geotherm profile,
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-[a comparison of the different **FD**-schemes applied on a 1-D gaussian temperature anomaly](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/DiffusionEquation/1D/Heat_1D_discretization.jl),
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-[a 2-D resolution test for each **FD**-scheme using a gaussian temperature anomaly](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/DiffusionEquation/2D/Gaussian_Diffusion.jl), and
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-[a resolution test for a 2-D poisson problem](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/DiffusionEquation/2D/Poisson_ResTest.jl).
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### [Heat Diffusion Equation](./man/DiffMain.md)
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For more examples see the [example folder](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/DiffusionEquation/).
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The [exercises](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/) include solving
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- the 1-D diffusion equation using the [forward](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/02_1D_Heat_explicit.ipynb) and [backward](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/03_1D_Heat_implicit.ipynb) Euler methods,
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-[a 2-D poisson problem](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/04_2D_Diffusion_Stationary.ipynb), and
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- a time-dependent temperature distribution within the lithosphere assuming a [plume](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/05_2D_Diffusion_TD_Plume.ipynb) or [sill](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/05_2D_Diffusion_TD_Sill.ipynb).
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The ```GeoModBox.jl``` provides different finite difference (**FD**) schemes to solve the *diffusive part* of the time-dependent or steady-state *temperature equation* including radioactive heating, in [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/). So far, only *Dirichlet* and *Neumann* thermal boundary conditions are available. Most of the functions assume constant thermal parameters (except for the 1-D solvers and the 2-D defect correction solver).
To solve the *advective part* of the *temperature equation*, the ```GeoModBox.jl``` provides the following different methods:
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- an upwind scheme,
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- a semi-lagrangian advection, and
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- passive tracers/markers.
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The solvers for a the tracer advection method are located in [src/Tracers](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/Tracers/), where the remaining advection routines are located in [src/AdvectionEquation](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/AdvectionEquation/). The routines are structured in such a way, that any property, as long as the property is defined on the *centroids* including *ghost nodes* on all boundaries, can be advected with the first **three** advection methods listed above. Using passive tracers, one can, so far, choose to either advect the temperature or phases. In case of advecting phases, one can define a certain rheology ($\eta$) or density ($\rho$) associated to each phase. The phase ID is used to interpolate the corresponding property from the tracers to the *centroids*.
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A key aspect for the advection equation is the conservation of the **amplitude** and the **shape** of an anomaly. Depending on the method, numerical diffusion or interpolation effects lead to strong deviations of the initial anomaly. For more details see [here](./man/AdvectMain.md). The ```GeoModBox.jl``` contains several [routines](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/InitialCondition/2Dini.jl) to setup a certain initial anomaly, either for properties defined on their correspondig grid (i.e., temperature, velocity, or phase) or for tracers advecting (so far) a certain temperature or phase. Within the examples and the exercise one can choose different initial temperature and velocity conditions.
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The examples for a two dimensional advection problem include:
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-[a 2-D advection, assuming a constant velocity field (e.g., a rigid body rotation)](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/AdvectionEquation/2D_Advection.jl), and
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-[a resolution test of the same advection example](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/examples/AdvectionEquation/2D_Advection_ResolutionTest.jl).
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The exercises include a:
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-[1-D advection of a gaussian or block anomaly](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/06_1D_Advection.ipynb), and
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-[a 2-D advection coupled with the solution of the diffusion equation](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/exercises/07_2D_Energy_Equation.ipynb).
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The ```GeoModBox.jl``` provides different methods to advect certain properties within the model domain. The corresponding routines are structured in such a way, that any property can be advected with the described advection solvers, as long as the property is defined on the *centroids* including *ghost nodes* at all boundaries. Using passive tracers, one can, so far, choose to either advect the absolute temperature or the phase ID.
**Figure 1. Gaussian Diffusion.** Time-dependent, diffusive solution of a 2-D Gaussian temperature anomaly using the [Crank-Nicholson approach](https://github.com/LukasFuchs/GeoModBox.jl/blob/main/src/HeatEquation/CNA.jl) in comparison to its analytical solution. Top Left: 2-D temperature field of the numerical solution and isotherms lines of the numerical (solid black) and analytical (dashed yellow) solution. Top Right: Total deviation to the analytical solution. Bottom Left: 1-D y-profile along x=0. Bottom Right: Root Mean Square total devation of the temperature over time.
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**Figure 1. Gaussian Diffusion.** Time-dependent, diffusive solution of a 2-D Gaussian temperature anomaly using the [Crank-Nicholson approach](https://github.com/GeoSci-FFM/GeoModBox.jl/blob/main/src/HeatEquation/CNA.jl) in comparison to its analytical solution. Top Left: 2-D temperature field of the numerical solution and isotherms lines of the numerical (solid black) and analytical (dashed yellow) solution. Top Right: Total deviation to the analytical solution. Bottom Left: 1-D y-profile along x=0. Bottom Right: Root Mean Square total devation of the temperature over time.
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**Figure 2. Resolution test.** Maximum *RMS* $\varepsilon$, maximum, and mean temperature for each **FD**-scheme and multiple resolutions.
**Figure 3. Rigid-Body-Rotation.** Time-dependent solution of a rotating circular temperature anomaly using the **upwind (first)**, **semi-lagrangian (second)**, and **tracer (third)** method. Within a circular area of our model domain the velocity is set to the velocity of a rigid rotation and outside euqal to zero. The temperature is scaled by the maximum temperature of the anomaly.
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