Merge pull request #6159 from gassmoeller/update_spiegelman_newton_be…

…nchmark

Update spiegelman newton benchmark

benchmarks/newton_solver_benchmark_set/spiegelman_et_al_2016/README.md

@@ -1,11 +1,19 @@
 # Newton Solver Benchmark Set - Spiegelman et at. (2016)
 
 The files in [this directory](https://github.com/geodynamics/aspect/tree/main/benchmarks/newton_solver_benchmark_set/spiegelman_et_al_2016)
-can be used to recreate the Spiegelman et al. (2016) flow figures from
-{cite:t}`fraters:etal:2019`. Adjust both the `metabash.sh` and `bash.sh` files to
-reflect the parameter search you are interested in and run the metabash script.
-If you have run exactly the same runs as shown in the paper, you can use the
-`cp_P-minLT-res-it-vel_BV.sh` script to recreate the figures in that paper.
-This script requires that the drucker prager compositions plugin is installed.
-It should be possible to replicate the same behavior with the visco plastic
-plugin.
+can be used to recreate the figures from
+{cite:t}`fraters:etal:2019` reproducing the {cite:t}`spiegelman:etal:2016` benchmark.
+`input.prm` is the setup of one of the benchmark
+models and `run.sh` will execute a series of model runs. Before you can execute
+the benchmark you will have to compile the plugin `drucker_prager_compositions.cc` in
+the same directory, the process of compiling plugins is described in [](sec:benchmark-run).
+After you have executed `run.sh`, you can use the gnuplot plotting script
+`plot.gnuplot` to recreate Fig. 4 of {cite:t}`fraters:etal:2019`.
+
+```{figure-md} fig:benchmark-newton-spiegelman-2016
+<img src="doc/figure_4.png" />
+
+Convergence history of several models of the Spiegelman et al. benchmark: a reproduction of three of the nine pressure dependent Drucker–Prager cases with a resolution of 64 × 16 cells. Top: results for computations where linear systems are solved with a maximum relative tolerance of 0.9. Bottom: The same models solved with a tolerance of $10^{-8}$. The initial Picard iteration is always solved to a linear tolerance of $10^{-14}$. Left to right: different prescribed velocities of u0 = 2.5, 5, and 12.5 cm\yr, and different background reference viscosities of respectively $\eta_{ref}$ = $10^{23}$, $10^{24}$ and $10^{25}$ Pa s. Horizontal axis: number of the non-linear (outer) iteration; and vertical axis: non-linear residual. "DC Picard" refers to a defect correction Picard iteration, see the paper describing this solver.
+```
+
+The nonlinear convergence behavior of ASPECT's different nonlinear solvers is shown in {numref}`fig:benchmark-newton-spiegelman-2016`.