This exercise will make you familiar the program sequence in DuMux and how different levels of complexity can be realized in the main file according to the complexity of your physical problem.

In order to do so, there are three examples of one phase flow problems. Two examples (a and b) are stationary problems and the third example (c) is an instationary problem.

The stationary examples differ in the fluidssystems they are using which means they differ in the fluid properties (e.g. density, thermal conductivity etc). The first problem (a) uses an incompressible fluid which means that the density does not change when pressure changes. This makes is possible to solve the system linearly. The second problem uses a compressible fluid, that means the density is a function of pressure and we need to use a nonlinear solver.

To summarize the problems differ in:

* exercise1_1p_a: a one-phase incompressible, stationary problem

* exercise1_1p_b: a one-phase compressible, stationary problem

* exercise1_1p_c: a one-phase compressible, instationary problem

The problem set-up for all three examples is always the same: It is a two dimensional problem and the domain is $`1m x 1m`$. It is a heterogeneous set-up with a lens in the middle of the domain which has a lower permeability ($`1e-12 m^2`$ compared to $`1e-10 m^2`$ in the rest of the domain).

In the beginning there is a uniform pressure of $`1e5 Pa`$ in the whole domain. On the top and the bottom border dirichlet boundary conditions are set with a pressure of $`1e5 Pa`$ on top and $`2e5 Pa`$ on the bottom. At the sides there is no in- or outflow and there are no source terms.

## Preparing the exercise

* Navigate to the directory `dumux/tutorial/ex1`

<br><br>

### Task 1: Getting familiar with the code

<hr>

Locate all the files you will need for this exercise

* The __main file__ for the __1p incompressible, stationary__ problem : `exercise1_1p_a.cc`

* The __main file__ for the __1p compressible, stationary__ problem : `exercise1_1p_b.cc`

* The __main file__ for the __1p compressible, instationary__ problem : `exercise1_1p_c.cc`

* The shared __problem file__: `1pproblem.hh`

* The shared __spatial parameters file__: `1pspatialparams.hh`

* The __input file__ for the __1p incompressible, stationary__ problem: `exercise1_1p_a.input`

* The __input file__ for the __1p compressible, stationary__ problem: `exercise1_1p_b.input`

* The __input file__ for the __1p compressible, instationary__ problem: `exercise1_1p_c.input`

Please pay special attention to the similarities and differences in the three main files. The first main file is solved linearly and does not need a newton solver or any other nonlinear solver method. The second problem is a nonlinear problem and uses newton's method to solve the system. The third problem is nonlinear and additionally instationary. Therefore a time loop needs to be included in the main file.

The general structure of any main file in DuMux is:

* the specific problem TypeTag is defined for the problem. This example shows the TypeTag for the CompressibleProblem

```c++

// define the type tag for this problem

usingTypeTag=TTAG(OnePCompressible);

```

* a gridmanager tries to create the grid either from a grid file or the input file

* we create the finite volume grid geometry, the problem, solutionvector and the gridvariables and initialize them. Additionally we initialize the vtkoutput. Each model has a predefined model specific output with relevant parameters for that model.

VtkOutputFields::init(vtkWriter);//!< Add model specific output fields

vtkWriter.write(0.0);

```

* then we need to assemble and solve the system. Depending on the problem this can be done with a linear solver or with a nonlinear solver. If the problem is time dependent we additionally need a time loop. An example for that is given in exercise1_1p_c:

* Compile all three executables `exercise1_1p_a` and `exercise1_1p_b` and `exercise1_1p_c`

```bash

make exercise1_1p_a exercise1_1p_b exercise1_1p_c

```

* Execute the three problems and inspect the result

```bash

./exercise1_1p_a

./exercise1_1p_b

./exercise1_1p_c

```

* you can look at the results with paraview

```bash

paraview injection-1p_a.pvd

```

<hr><br><br>

### Task 3: Analytical differentiation

<hr>

For the incompressible one phase problem it is possible to also have an analytic solution method. To implement that follow the directions in the `exercise1_1p_a.cc` and the `1pproblem.hh` marked by: