# Solvers ## [Saint-Venant](saint-venant.h) $$\partial_t \int_{\Omega} \mathbf{q} d \Omega = \int_{\partial \Omega} \mathbf{f} ( \mathbf{q}) \cdot \mathbf{n}d \partial \Omega - \int_{\Omega} hg \nabla z_b$$ $$\mathbf{q} = \left(\begin{array}{c} h\\ hu_x\\ hu_y \end{array}\right), \;\;\;\;\;\; \mathbf{f} (\mathbf{q}) = \left(\begin{array}{cc} hu_x & hu_y\\ hu_x^2 + \frac{1}{2} gh^2 & hu_xu_y\\ hu_xu_y & hu_y^2 + \frac{1}{2} gh^2 \end{array}\right)$$ * [Semi-implicit scheme](saint-venant-implicit.h) * [Multiple layers](multilayer.h) $$\partial_th + \partial_x\sum_{l=0}^{nl-1}h_lu_l = 0$$ $$\partial_t(h\mathbf{u}_l) + \nabla\cdot\left(h\mathbf{u}_l\otimes\mathbf{u}_l + \frac{gh^2}{2}\mathbf{I}\right) = - gh\nabla z_b - \partial_z(h\mathbf{u}w) + \nu h\partial_{z^2}\mathbf{u}$$ * [Green-Naghdi](green-naghdi.h) $$\partial_t \int_{\Omega} \mathbf{q} d \Omega = \int_{\partial \Omega} \mathbf{f} ( \mathbf{q}) \cdot \mathbf{n}d \partial \Omega - \int_{\Omega} hg \nabla z_b + h \left( \frac{g}{\alpha}\nabla \eta - D \right)$$ $$\alpha h\mathcal{T} \left( D \right) + hD = b$$ $$b = \left[ \frac{g}{\alpha} \nabla \eta +\mathcal{Q}_1 \left( u \right) \right]$$ ## [Systems of conservation laws](conservation.h) $$\partial_t\left(\begin{array}{c} s_i\\ \mathbf{v}_j\\ \end{array}\right) + \nabla\cdot\left(\begin{array}{c} \mathbf{F}_i\\ \mathbf{T}_j\\ \end{array}\right) = 0$$ * [Compressible gas dynamics](compressible.h) * [All Mach compressible flows](all-mach.h) $$\partial_t\mathbf{q} + \nabla\cdot(\mathbf{q}\mathbf{u}) = - \nabla p + \nabla\cdot(\mu\nabla\mathbf{u}) + \rho\mathbf{a}$$ $$\partial_t p + \mathbf{u}\cdot\nabla p = -\rho c^2\nabla\cdot\mathbf{u}$$ ## Navier--Stokes * [Streamfunction--Vorticity formulation](navier-stokes/stream.h) $$\partial_t\omega + \mathbf{u}\cdot\nabla\omega = \nu\nabla^2\omega$$ $$\nabla^2\psi = \omega$$ * ["Markers-And-Cells" (MAC or "C-grid") formulation](navier-stokes/mac.h) * [Centered formulation](navier-stokes/centered.h) $$\partial_t\mathbf{u}+\nabla\cdot(\mathbf{u}\otimes\mathbf{u}) = \frac{1}{\rho}\left[-\nabla p + \nabla\cdot(2\mu\mathbf{D})\right] + \mathbf{a}$$ $$\nabla\cdot\mathbf{u} = 0$$ + [Azimuthal velocity for axisymmetric flows](navier-stokes/swirl.h) $$\partial_t w + u_x \partial_x w + u_y \partial_y w + \frac{u_y w}{y} = \frac{1}{\rho y} \left[ \nabla \cdot (\mu y \nabla w) - w \left( \frac{\mu}{y} + \partial_y \mu \right) \right]$$ * [Two-phase interfacial flows](two-phase.h) + [Momentum conservation option](navier-stokes/conserving.h) * [Momentum-conserving two-phase interfacial flows](momentum.h) ## Electrohydrodynamics * [Ohmic conduction](ehd/implicit.h) $$\partial_t\rho_e = \nabla \cdot(K \nabla \phi)$$ $$\nabla \cdot (\epsilon \nabla \phi) = - \rho_e$$ * [Ohmic conduction of charged species](ehd/pnp.h) $$\partial_tc_i = \nabla \cdot( K_i c_i \nabla \phi)$$ * [Electrohydrodynamic stresses](ehd/stress.h) $$M_{ij} = \varepsilon (E_i E_j - \frac{E^2}{2}\delta_{ij})$$ ## Viscoelasticity * [The log-conformation method for some viscoelastic constitutive models](log-conform.h) $$\rho\left[\partial_t\mathbf{u}+\nabla\cdot(\mathbf{u}\otimes\mathbf{u})\right] = - \nabla p + \nabla\cdot(2\mu_s\mathbf{D}) + \nabla\cdot\mathbf{\tau}_p + \rho\mathbf{a}$$ $$\mathbf{\tau}_p = \frac{\mu_p \mathbf{f_s}(\mathbf{A})}{\lambda}$$ $$\Psi = \log\mathbf{A}$$ $$D_t \Psi = (\Omega \cdot \Psi -\Psi \cdot \Omega) + 2 \mathbf{B} + \frac{e^{-\Psi} \mathbf{f}_r (e^{\Psi})}{\lambda}$$ ## Other equations * [Hele-Shaw/Darcy flows](hele-shaw.h) $$\mathbf{u} = \beta\nabla p$$ $$\nabla\cdot(\beta\nabla p) = \zeta$$ * [Advection](advection.h) $$\partial_tf_i+\mathbf{u}\cdot\nabla f_i=0$$ + [Volume-Of-Fluid advection](vof.h) + [Contact angles](contact.h) * [Interfacial forces](iforce.h) $$\phi\mathbf{n}\delta_s$$ + [Surface tension](tension.h) + [Reduced gravity](reduced.h) * [Reaction--Diffusion](diffusion.h) $$\theta\partial_tf = \nabla\cdot(D\nabla f) + \beta f + r$$ * [Poisson--Helmholtz](poisson.h) $$\nabla\cdot (\alpha\nabla a) + \lambda a = b$$ * [Runge--Kutta time integrators](runge-kutta.h) $$\frac{\partial\mathbf{u}}{\partial t} = L(\mathbf{u}, t)$$ * [Signed distance field](distance.h) * [Okada fault model](okada.h) # General orthogonal coordinates When not written in vector form, some of the equations above will change depending on the choice of coordinate system (e.g. polar rather than Cartesian coordinates). In addition, extra terms can appear due to the geometric curvature of space (e.g. equations on the sphere). An important simplification is to consider only [orthogonal coordinates](http://en.wikipedia.org/wiki/Orthogonal_coordinates). In this case, consistent finite-volume discretisations of standard operators (divergence etc...) can be obtained, for any orthogonal curvilinear coordinate system, using only a few additional geometric parameters. ![Metric scale factors](figures/metric.svg) The [face vector](/Basilisk C#face-and-vertex-fields) *fm* is the *scale factor* for the length of a face i.e. the physical length is $fm\Delta$ and the scalar field *cm* is the scale factor for the area of the cell i.e. the physical area is $cm\Delta^2$. By default, these fields are constant and unity (i.e. the Cartesian metric). Several metric spaces/coordinate systems are predefined: * [Axisymmetric](axi.h) + [Stokes stream function](axistream.h) $$\frac{\partial^2\psi}{\partial z^2} + \frac{\partial^2\psi}{\partial r^2} - \frac{1}{r}\partial_r\psi = - \omega r$$ * [Spherical](spherical.h) * [Radial/cylindrical](radial.h) # Data processing * [Various utility functions](utils.h): timing, field statistics, slope limiters, etc. * [Tagging connected neighborhoods](tag.h) * [Counting droplets](examples/atomisation.c#counting-droplets) # Output functions * [Multiple fields interpolated on a regular grid (text format)](output.h#output_field-multiple-fields-interpolated-on-a-regular-grid-text-format) * [Single field interpolated on a regular grid (binary format)](output.h#output_matrix-single-field-interpolated-on-a-regular-grid-binary-format) * [Portable PixMap (PPM) image output](output.h#output_ppm-portable-pixmap-ppm-image-output) * [Volume-Of-Fluid facets](fractions.h#interface-output) * [Basilisk snapshots](output.h#dump-basilisk-snapshots) * [Basilisk View](view.h) * [Gerris simulation format](output.h#output_gfs-gerris-simulation-format) * [ESRI ASCII Grid format](output.h#output_grd-esri-ascii-grid-format) * [VTK format](vtk.h) # Input functions * [Basilisk snapshots](output.h#dump-basilisk-snapshots) * [Gerris simulation format](input.h#input_gfs-gerris-simulation-format) * [ESRI ASCII Grid format](input.h#input_grd-raster-format-esri-grid) * [Portable Gray Map (PGM) images](input.h#input_pgm-importing-portable-gray-map-pgm-images) # Interactive Basilisk View * [bview](bview): a script to start the client/server visualisation pipeline. * [bview-server.c](bview-server.c): the server. * [bview-client.py](bview-client.py): the client. # Miscellaneous functions/modules * [Controlling the maximum runtime](maxruntime.h) # Tracking floating-point exceptions On systems which support [signaling NaNs](http://en.wikipedia.org/wiki/NaN#Signaling_NaN) (such as GNU/Linux), Basilisk is set up so that trying to use an unitialised value will cause a floating-point exception to be triggered and the program to abort. This is particularly useful when developing adaptive algorithms and/or debugging boundary conditions. To maximise the "debugging potential" of this approach it is also recommended to use the trash() function to reset any field prior to updates. This will guarantee that older values are not mistakenly reused. Note that this call is quite expensive and needs to be turned on by adding -DTRASH=1 to the compilation flags (otherwise it is just ignored). Doing ~~~bash ulimit -c unlimited ~~~ before running the code will allow generation of core files which can be used for post-mortem debugging (e.g. with gdb). ## Visualising stencils It is often useful to visualise the values of fields in the stencil which triggered the exception. This can be done using the -catch option of qcc. We will take this code as an example: ~~~c #include "utils.h" int main() { init_grid (16); scalar a[]; trash ({a}); foreach() a[] = x; vector ga[]; gradients ({a}, {ga}); } ~~~ Copy and paste this into test.c, then do ~~~bash ulimit -c unlimited qcc -DTRASH=1 -g -Wall test.c -o test -lm ./test ~~~ you should get ~~~ Floating point exception (core dumped) ~~~ Then do ~~~bash gdb test core ~~~ you should get ~~~ ... Core was generated by ./test'. Program terminated with signal 8, Arithmetic exception. #0 0x0000000000419dbe in gradients (f=0x7fff5f412430, g=0x7fff5f412420) at /home/popinet/basilisk/wiki/src/utils.h:203 203 v.x[] = (s[1,0] - s[-1,0])/(2.*Delta); ~~~ i.e. the exception occured in the [gradients()](utils.h#gradients) function of [utils.h](). To visualise the stencil/fields which lead to the exception do ~~~bash qcc -catch -g -Wall test.c -o test -lm ./test ~~~ you should now get ~~~ Caught signal 8 (Floating Point Exception) Caught signal 6 (Aborted) Last point stencils can be displayed using (in gnuplot) set size ratio -1 set key outside v=0 plot 'cells' w l lc 0, \ 'stencil' u 1+3*v:2+3*v:3+3*v w labels tc lt 1 title columnhead(3+3*v), \ 'coarse' u 1+3*v:2+3*v:3+3*v w labels tc lt 3 t '' Aborted (core dumped) ~~~ Follow the instructions i.e. ~~~bash gnuplot gnuplot> set size ratio -1 gnuplot> set key outside gnuplot> v=0 gnuplot> plot 'cells' w l lc 0, \ 'stencil' u 1+3*v:2+3*v:3+3*v w labels tc lt 1 title columnhead(3+3*v), \ 'coarse' u 1+3*v:2+3*v:3+3*v w labels tc lt 3 t '' ~~~ With some zooming and panning, you should get this picture ![Example of stencil/field causing an exception](figures/catch.png) The red numbers represent the stencil the code was working on when the exception occured. It is centered on the top-left corner of the domain. Cells both inside the domain and outside (i.e. ghost cells) are represented. While the field inside the domain has been initialised, ghost cell values have not. This causes the gradients() function to generate the exception when it tries to access ghost cell values. To initialise the ghost-cell values, we need to apply the boundary conditions i.e. add ~~~c boundary ({a}); ~~~ after initialisation. Recompiling and re-running confirms that this fixes the problem. Note that the blue numbers are the field values for the parent cells (in the quadtree hierarchy). We can see that these are also un-initialised but this is not a problem since we don't use them in this example. The v value in the gnuplot script is important. It controls which field is displayed. v=0 indicates the first field allocated by the program (i.e. a[] in this example), accordingly ga.x[] and ga.y[]` have indices 1 and 2 respectively. ## Tracing permissions Some recent systems disallow tracing of processes for security reasons. The symptom will be an error message from gdb looking like: ~~~bash Attaching to process 9351 Could not attach to process. If your uid matches the uid of the target process, check the setting of /proc/sys/kernel/yama/ptrace_scope, or try again as the root user. For more details, see /etc/sysctl.d/10-ptrace.conf ptrace: Operation not permitted. ~~~ To enable tracing (which weakens your system's security), you need to do: ~~~bash sudo sh -c 'echo 0 > /proc/sys/kernel/yama/ptrace_scope' ~~~ # See also * [Tips]() * [Built-in profiling](README.trace) * [Built-in memory profiling](README.mtrace) * [Performance profiling with Paraver](README.paraver) * [Profiling round-off errors with CADNA](README.cadna)