FVSYST 1.0 Manual
Contents
1. fc 2 1c w a u n u bw 2 1d with the homogeneous Neumann boundary conditions Vu n 0 on OD x 0 T Vaen 0 on Q x 0 7 Ve n 0 on OO x 0 T Vw n 0 on Q x 0 7 and with the initial conditions u 0 uo i 6 n 0 no in Q a0 in Q w 0 wo in 2 Here u u x t n n x t c c x t and w w x t are the unknown functions representing respectively the density of active cells nutrient concentration attractant concentration and the density of inactive cells Whereas du dn de a B y b and k are scalar parameters 0 if u lt Ucrit gon if u gt Ueit glu n Here go and Ucrit are again scalar parameters The two following variants of the function y are considered either c x c c 1 or g c MO ay where 0 is a scalar parameter Finally for the function a there are three choices ao a u n EN Le u n S alun _ G alun a Here again ao a1 and ag are scalar parameters 3 Numerical scheme The finite volume method cf 2 is used for the discretization Details are to come 4 Using FVSYST FVSYST is launched from the Matlab command line using the coefficients and parameters defined in the file FVSYST m see Sections 4 1 and 4 2 for their specification Before starting the actual execution please type clear first this command erases all variables actually existing in the Matlab workspace When loading geometry data from a2. N_vis_sv toot Wo Or vis_sv_0 1 if N_eq 3 dep_table end if N_eq dep_table PES RAR 1 1 0 1 0 1 iy tg Ay 2 h h 1 2 or 3 type of the function a 1 or 2 type of the function xi number of equations actually in the system to compute time interval is O T basic length of a time step type of solution 1 5 see above O or 1 whether to visualise the results on the screen which unkn to visualize 0 all or 1 4 O or 1 whether to create a video avi if vis 1 O or 1 whether to save jpg files when vis 1 O or 1 whether to save the results to mat files no of figures to be vis saved or added to video saved to a file O if none vis save or add to video save to a file init conds dependence between the individual unknowns end epsilon le 2 NZERO ie 15 scale 20 type_IC 2 rad 0 008 exec_save_geom geom_file_name geom_creat 0 rf_msh 0 N_lay 5 if geom_creat 1 gt geom_999 0 load geom_78080_quart load geom_16256_ref5 load geom_16256 load geom_254 end epsilon approx by a smooth function of g rounding error tolerance scaling factor of the initial geometry type of the initial condition for the first unknown 4 O the element containing the point 0 0 1 the element containing 0 0 and its neighbors 2 all elements with bar closer to 0 0 than rad radius for type_IC whether to execute 1 or to
3. in order to run a simulation first type in MATLAB clear clear variables from MATLAB workspace clc clear MATLAB command line pdecirc 0 0 1 Domain domain generation using pdetool new window opens here do Mesh gt Parameters Mesh gt Initialize Mesh Mesh gt Export Mesh now you have in MATLAB workspace a Delaunay triangular mesh 4 p points e boundary edges t triangles the last steps can be replaced by reading externally the geometry see below set all the parameters below in this file and SAVE it back in MATLAB command line type FVSYST this launches the simulation global N_el N_eq areas dif_matr dif_matr_O_diag conv_matr g_handle global a_handle xi_der_handle cf_dif cf_u_crit cf_g_0O epsilon global type_fn_a cf_a_O cf_a_1 cf_a_2 cf_alpha cf_beta cf_gama global cf_b type_fn_xi cf_theta cf_k xi_handle Ahhhhhhhhhhhhhhhhhhhhhh PARAMETERS DEFINITION Ahhhhhhhhhhhhhhhhhhhhhh cf_dif 1 0 01 cf_dif 2 0 02 cf_dif 3 0 1 cf_dif 4 0 cf_alpha 1 cf_beta 1 cf_gama 1 cf_b 0 cf_u_crit 0 05 cf_g_0 1 type_fn_a 3 type_fn_xi 2 cf_a_O 0 05 cf_a_1 0 1 cf_a_2 1 cf_theta 0 25 cf_k 0 053 init 1 init 2 init 3 init 4 oul oOOrRF we Ahhhhhhhhhhhhhhhhhhhhh VARIABLES DEFINITION Ahhhhhhhhhhhhhhhhhhhhh N_eq 3 TIME 900 dt 1 sol_type 2 vis 1 vis_unkn 0 video 1 save_figs 0 save_res
4. 1 th step of the Newton method Then assembling the Newton matrix of this approximate solution is usually sufficient to move the approximation towards the exact solution of the original nonlinear matrix problem 4 2 6 Using preconditioning We strongly recommend to use preconditioning and permutation for the solution of linear systems i e set precond_perm 1 when sol_type is 1 2 or 3 The example in Appendix C shows the influence this setting has in the first case no preconditioning is used and the iterative solver GMRES in this case is applied directly in the incomplete Newton method We can see that the time step had to be cut and yet the convergence is very slow The total solution cost is uncomparable with that where preconditioning and permutation were used see the second case 4 2 7 LU factorization only at the first step In the preconditioning technique it is sufficient that the obtained L and U matrices represent only approximate LU decomposition of the matrix at hand What is important is finally the condition number of the preconditioned matrix Hence it may not be necessary to decompose the matrix at hand always if we already have some decomposition So that in particular for the nonlinear problems at hand we may only make the LU factorization on the first linearization step and then keep the obtained L and U factors also for subsequent Newton linearization steps This usually leads to important savings in CPU time and
5. In particular it has been created in order to simulate the growth of and pattern formation by colonies of the bacteria Escherichia coli observed by Budrene and Berg 1 It is based on the finite volume method cf 2 for the discretization in space and futures implicit or explicit options concerning the time discretization In the implicit case preconditioned iterative methods and inexact version of the Newton method are used for the solution of the arising systems of nonlinear algebraic equations In this case adaptive time step cutting is also used when the Newton method does not converge fast enough FVSYST is a Matlab program and it has been run on Windows Unix as well as Apple Mac operating systems The code is highly optimized so that it allows to consider problems with more than one million spatial unknowns and several hundreds time steps even on a personal computer or a laptop Finally different visualization features are implemented such that on screen visualization video avi files creation or jpg files graphical output saving An example of a simulated pattern is given in Figure 2 in Appendix B below 2 Domain and equations Let Q C R be a polygonal domain open bounded and connected set with boundary 09 and let 0 T 0 lt T lt be a time interval At the present time the following system is considered in FVSYST uw dAu yg u n u kV uVx c a u n u bw 2 1a m d An g u n u 2 1b a dAc au
6. we thus suggest to put LU_fact_only_first 1 when precond_perm 1 and sol_type is 1 2 or 3 References 1 BUDRENE E O AND BERG H C Dynamics of formation of symmetrical patterns by chemotactic bacteria Nature 376 6535 1995 49 53 2 EYMARD R GALLOUET T AND HERBIN R Finite volume methods In Handbook of Numerical Analysis Vol VIT North Holland Amsterdam 2000 pp 713 1020 3 QUARTERONI A SACCO R AND SALERI F Numerical mathematics vol 37 of Texts in Applied Mathematics Springer Verlag New York 2000 A Example of a control file a A FVSYST m main file for convection reaction diffusion system simulation model finite volume method on strict Delaunay triangular grids the sum of oposite angles for each edge has to be less than pi five versions fully implicit with full Newton method used for the linearization fully implicit with full Newton with the exception of the convection term explicit with respect to convection implicit and Newton otherwise s e UNBE completely explicit in Matlab 5 completely explicit in C preconditioned iterative methods and inexact version of the Newton method 4 used for the solution of linear systems versions 1 3 adaptive time step cutting when Newton converges slowly versions 1 3 uses standard MATLAB functions plus the PDE toolbox for meshes generation based on MATLAB 6 1 Martin Vohralik September 2006 hhhhhhh INTRO hhhhhhh
7. 00615e 003 15573e 004 25456e 004 N OH EH N A 12034e 002 83608 TIME 6 06 23 11 28 02 19 30 53 Oo Oo OO OO OO Oo OD 12 OO OM OO Oe 00 00 00 00 00 00 00 00 00 20 20 20 20 20 20 20 20 20 OO OO OM OO Oe ONwWwoPRPON OF 38157e 003 39214e 004 24935e 004 94309e 004 99147e 004 27896e 004 73084e 004 29942e 004 94712e 004 Newton step Newton linearization error matrix and rhs assemblage time LUinc time and number of iterations relative residual and CPU time of the iterative method 1 5 50001e 002 6 27 31 04 12 7 0 8 65547e 009 16 16 16 17 16 16 17 19 16 16 16 17 16 16 16 19 19 19 12 00 31 23 28 32 00 20 23 00 14 13 50 05 82 13 79 99 2 3 64712e 003 6 94 3 1 60441e 004 6 93 4 9 84356e 006 6 76 5 7 01058e 007 7 17 TIME 0 2 0 00 0 00 0 00 0 00 o gt o gt Or e gt O1 O1 O1 o 7 28095e 009 7 31255e 009 7 31458e 009 7 31473e 009 Newton step Newton linearization error matrix and rhs assemblage time LUinc time and number of iterations relative residual and CPU time of the iterative method 1 3 43968e 002 6 57 2 6 26778e 003 6 61 3 4 87399e 004 6 64 4 4 82504e 005 6 75 30 43 0 00 0 00 0 00 13 7 7 T 6 5 0 0 0 8 13132e 009 3 74897e 009 3 82000e 009 3 83493e 009 40 45 16 02 15 29 1
8. 5 18 14 86 16 60 16 01 16 10
9. FVSYST 10 A Matlab finite volume code for simulation of nonlinear convection reaction diffusion systems in the life sciences User Manual Martin Vohralik October 1 2006 1 Laboratoire Jacques Louis Lions Universit Pierre et Marie Curie Paris 6 175 rue du Chevaleret 75 013 Paris France vohralik ann jussieu fr Contents 1 Purpose and scope 3 2 Domain and equations 3 3 Numerical scheme 4 4 Using FVSYST 4 1 Specifying FVSYST coefficients 4 2 Specifying FVSYST parameters soa 4 4 opus oh nus 4 2 1 Precise form of the equation system to be solved 42 2 Choosing type of the numerical solution 42 3 Visualization video or image files creation and results saving AQA Mesh specification 5 4 44 vue ns du ds ee Mu hu 4 2 5 Inexact Newton method 476 Using preconditioning ecs sw acs a a ha ana dus ee R 4h 4 2 7 LU factorization only at the first step OO EE BREA A Example of a control file 8 Example of a bacteria pattern 11 C Example of the influence of the use of preconditioning for the solution of linear systems 11 1 Purpose and scope FVSYST is a program for numerical simulation of systems of nonlinear convection reaction diffusion equations in two dimensional domains like those describing various processes in the life sciences
10. ations are to be solved but quite severe maximal time step condition is necessary imposing a huge number of arithmetic operations The choice 5 should be faster than 4 by a factor of about 2 4 2 8 Visualization video or image files creation and results saving Visualization in FVSYST is turned on by setting vis 1 In parameter vis_unkn on should then specify whether one wants to visualize the results individually u n c or u w val ues 1 4 in the corresponding order or whether one graph with all four results should be given vis_unkn 0 By setting save_res to 0 or 1 we can decide whether the results will be saved The parameter N_vis_sv then specifies the number of times when the results should be visualized and or saved to the disk Independently the initial condition may be visualized saved to disk by setting vis_sv_0 1 Finally whether the video file video avi is created is decided by the parameter video and similarly jpg files are committed to the disk by putting save_figs 1 4 2 4 Mesh specification There are two basic ways to specify a mesh in FVSYST If there already is some mesh created and stored in a file we put geom_creat 0 and specify the name of the file with the mesh e g geom_254 mat In the other case which corresponds to geom_creat 1 we can create a mesh with the PDE Toolbox as explained at the beginning of Section 4 To save a newly created mesh actually not only the mesh but some other variable
11. file see Section 4 2 4 there is nothing more to do and you can start the execution by typing FVSYST on the Matlab command line and pressing enter When you want to create a new mesh open first the PDE Toolbox specify there the domain and the mesh and then export the vertices of the mesh p the edges e and the triangles t to the Matlab workspace Only then type FVSYST 4 1 Specifying FVSYST coefficients All the coefficients dy dn de a B Y b k go Ucrit 9 ao a1 a2 as well as the choice of the functions x and a and the values of initial conditions are specified in the file FVSYST m Do not forget to save the file FVSYST m each time you make some change 4 2 Specifying FVSYST parameters All FVSYST parameters are to be specified in the second part of the file FVSYST m You do not need to and should not change anything else The majority of the settings are sufficiently explained in the file FVSYST m itself see Appendix A for its example so that we only highlight some particularities 4 2 1 Precise form of the equation system to be solved In the actual form of 2 1a 2 1d the coefficient b may be equal to zero In this case the equa tion 2 1d is in fact decoupled from the system 2 1a 2 1c Specifying N_eq 3 in FVSYST m this decoupled form will be used whereas letting N_eq 4 the four equation system will be solved The choice N_eq 3 is recommended as long as b is zero 4 2 2 Choosing type of the numerical sol
12. s as well to a file set geom_creat 1 and exec_save_geom 0 and specify the geom_file_name The execution of FVSYST now ends after saving your mesh and you then have to put exec_save_geom 1 SAVE FVSYST m and start over again If geom_creat 1 there is an option rf_msh enabling to refine the mesh into N_lay layers around the origin cf Figure 1 E ARION gt RSR SRE 7 KL A NIRE NSA K I La 4 AA WV A VA TRES KT EERE Re RS 7 RIDER NA NS 7 LIN IS CREE IOOO YO SRE 0 EX AVANT en XA CAES LE INRP SSAA RISERS KA RAR ARR VXD Figure 1 A mesh refined into 3 layers 4 2 5 Inexact Newton method For nonlinear problems the inexact Newton method see e g 3 is a connection of the classical Newton method for the solution of a nonlinear matrix problem with an iterative method for the solution of resulting linear matrix problems In our case we apply it when sol_type is 1 2 or 3 One limits here the maximum number of iterations of the iterative solver max_it while not limiting severely the maximal number of iterations of the Newton linearization max_lin_it before cutting the time step Having an approximate solution on a k th step of the Newton method several iterations of the iterative method started from this solution do not give an exact result of the k 1 th step of the Newton method but are usually sufficient to move the approximate solution towards the exact solution of the k
13. save geometry files 0 file name to save geometry to if exec_save_geom 0 1 or 0 either create the arrays areas neigh dif_matr dif_matr_O_diag on the basis of p and t 4 exported to MATLAB from the PDE toolbox or read it from a disk saves time whether to refine the mesh when geom_creat number of refinement layers in this case file of areas neigh dif_matr dif_matr_O_diag p t variables to set for linear systems solution if applicable if sol_type shift 1e 8 err_lin_crit max_lin_it sol_type 1e 6 20 max_lin_it_dec 8 tol 1e 8 max_it 20 precond_perm 1 if precond_perm LU_fact_only_first 1 use LU incomplete prec only at the ist Newton step perm_type 1 end end FVSYST_exec sol_type 3 implicit or semi implicit scheme epsilon in the approx diff in the Newton method Newton stopping criterion relative L2 norm max no of Newton iter before cutting the time step as above maximum no of iterations for the relative 4 residual to decrease below err_lin_crit 1 2 accuracy of iterative method if used 4 max number of iter of iterative method if used may be very small in the inexact Newton method 0 or 1 whether to use prec and column permutation type of permutation for reordering of the equations 1 like as nmb first by elements then by equations 2 column minimum degree permutation 3 approximate column minim
14. um degree permutation 10 B Example of a bacteria pattern Approximation of active cells u Approximation of total cells u w 20 3 5 10 3 2 5 0 2 1 5 10 1 0 5 20 20 20 10 10 0 0 10 10 20 20 20 10 20 10 0 10 20 Figure 2 Escherichia coli pattern obtained with the settings of Section A at t 850 C Example of the influence of the use of preconditioning for the solution of linear systems gt gt clear gt gt FVSYST N_el 83608 11 TIME Newton step Newton linearization error matrix and rhs assemblage time LUinc time and number of iterations relative residual and CPU time of the iterative method 1 Oo O1 N TIME 3 26316e 003 13614e 003 40886e 003 23476e 003 77555e 003 62409e 003 32125e 003 RH H HO N NN 24705e 002 6 02 06 18 16 15 30 41 Oo Oo OO OO OO OD 46 OOOOOCO 00 00 00 00 00 00 00 00 20 20 20 20 20 20 20 20 OO OOOoOOCO HR H H H HA ED W 11146e 003 31623e 003 89483e 003 62256e 003 42212e 003 26875e 003 14997e 003 05534e 003 Newton step Newton linearization error matrix and rhs assemblage time LUinc time and number of iterations relative residual and CPU time of the iterative method 1 OMAN OO O1 FWD gt gt clear gt gt FVSYST N_el 3 16027e 003 37984e 003 86042e 003 52049e 003 20550e 003
15. ution In variable sol_type you specify the type of the solution technique to approximate a solution to 2 1a 2 1d The choices are 1 Fully implicit scheme in time with complete Newton method for the linearization 2 Fully implicit scheme in time where the Newton method is not applied to the convection term kV uVx c in 2 1a values of c from previous Newton step are used here 3 Explicit scheme in time with respect to convection implicit with respect to the other terms The Newton method is applied to the solution of the nonlinear system of algebraic equations 4 Fully explicit scheme in time All the computations are done in Matlab 5 Fully explicit scheme in time The bottleneck computations are done in a stand alone C code explicitc c compiled to an appropriate executable explicitc d11 in Windows The recommended choice best precision fastest execution is 2 Both choices 1 or 2 assure an unconditional stability of the scheme virtually unlimited maximal time steps but lead to a system of nonlinear algebraic equations to be solved on each time step They only differ in the solution of the resulting systems hopefully 2 is much faster than 1 In choice 3 there is a not too severe maximal time step condition the scheme is only conditionally stable but still nonlinear systems are to be solved Thus this choice will probably be slower than 2 In choices 4 and 5 no linear or nonlinear systems of algebraic equ
Download Pdf Manuals
Related Search