Dornic integration method for multipicative noise

---

Overview

Dornic_et_al_integration_method.h provides a general class to integrate stochastic differential equations.

It is an optimization and generalization of a basic code implemented by Ivan Dornic and Juan A. Bonachela based on the algorithm proposed in:

“Integration of langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions”; Ivan Dornic, Hugues Chaté, and Miguel A Munoz; Physical review letters 94; 100601 (2005).

and improved following:

“Simulation of spatial systems with demographic noise”; Haim Weissmann, Nadav M Shnerb, and David A Kessler; Physical Review E 98; 022131 (2018).

This project is licensed under the terms of the GNU General Public License license.

If you use our code in an academic publication, please include us in the acknowledgements.

---

Installation and compiling

Download or clone this repository using git clone.. Integration routines are inside the fileDornic_et_al_integration_method.h, that should included in your file. Then compile the code as usual,

g++ example_main.cpp -std=c++11 -other_flags -o example_main

---

Use

In order to show the proper way of using it, we implemented "example_main.cpp". These codes integrate the stochastic equation described in "Dornic_method.pdf" in a 1D lattice.

1. To integrate a stochastic equation with different non-linear terms you may need to change the functions:

    ```c++
   

   ///////////////////////// NON-LINEAR TERM INTEGRATION FUNCTIONS /////////////////////// void set_non_linear_coefficients(vector f_parameters){ quadratic_coefficient=(f_parameters)[3]; } double non_linear_term_integration(int inode, vector f_in, double dt_in){ return -quadratic_coefficient(cell_density[inode]+dt_in(f_in)[inode])(cell_density[inode]+dt_in(*f_in)[inode]); }

2. To integrate a stochastic equation with a different adyacency network, create a new function similar to the following:

   ///////////////////////// NETWORK ///////////////////////
   	void set_1D_lattice(){
   		double N=cell_density.size();
   		for(int inode=0;inode<N;inode++) nodes_neighbors_vector[inode].neighbors.clear();
   		for(int inode=1;inode<N-1;inode++){
   			nodes_neighbors_vector[inode].neighbors.push_back(inode-1);
   			nodes_neighbors_vector[inode].neighbors.push_back(inode+1);
   		}
   			nodes_neighbors_vector[0].neighbors.push_back(1);
   	nodes_neighbors_vector[N-1].neighbors.push_back(N-2);
   }

3. To integrate a differential equation with coefficients that change in time:

   In Dornic_et_al_integration_method.h: change #define CONSTANT_COEFFICIENTS false In main.cpp: use the method f_dornic.integrate(r,&f_parameters);

4. To integrate a differential equation with number of cells that change in time:

    In `Dornic_et_al_integration_method.h`:  make `#define CONSTANT_CELLS_NUMBER false`
   

---