To build Ordinary Differential Equations models (ODE)


The vle extension vle.ode allow to model and simulate ordinary differential equations (ode) into VLE. It provides three numerical methods for integration of ode:

  • Euler: referred below as a time slicing method
  • Runge-Kutta 4: referred below as a time slicing method
  • QSS2: quantized state systems of order 2; the implementation is the one proposed by Kofman in ‘A Second Order Approximation for DEVS Simulation of Continuous Systems’ (Simulation, 2002).
User interface of ODE models

Fig: The user interface of an atomic model of type ODE.

Evolution of state variables Vi is described by differential equations. These expressions can rely on the value of external variables Ei. For time slicing methods, external variables Ei are expected to be piecewise constant functions computed from continuous and derivable functions. For QSS2, external variables are expected to be piecewise linear functions.

Atomic model ports Ei and Vi can carry data at time t that contain:

  • ‘value’: this is the value at $t$ of the variable ‘name’.
  • ‘gradient’ (optional): this is the value of the gradient of the variable ‘name’ at t.

Writing ODE atomic models

Below is given an example of dynamic for an atomic model that relies on the vle.ode

class MyModel : public DifferentialEquation
 MyModel(const DynamicsInit& model,
	     const InitEventList& events) :
    //Initialization of variables is done 
    //into the class constructor:
    V1.init(this, "V1", events);
    E1.init(this, "E1", events)
 //gradients of state variables are expressed 
 //into the 'compute' function 
 void compute(const Time& time)
    grad(V1) = V1() - E1();
 //states and external variables are attributes of the class
 Var V1;
 Var E1;//considered here as external variable

Configuring ODE atomics models

Basic settings:

  • method (either ‘euler’, ‘rk4’ or ‘qss2’, default ‘euler’): it specifies the method used for numerical integration.
  • init_value_X (double, default 0.0): initial value of the variable X.
  • init_grad_X (double, default 0.0): initial first order derivative of the variable X.

Settings for time slicing methods methods (‘euler’ or ‘rk4’):

  • time_step (double, default 1.0) : the time step for the numerical integration.
  • output_period (uint, default 1): gives the time step of output. Output will produce values each time_step * output_period.

Settings for quantized methods (‘qss2’):

  • quantum_X (double, default 0): it gives the quantum for quantization of variable X. If it equals 0, then it is expected to be an external variable, then no quantization is performed on X.

Technical details

Description of the dynamic is based on DEVS state graph transition. For QSS2 implementation, the most general case is considered; ie. derivative expressions are non linear functions depending on all state variables and thus:

  • gradient derivatives are computed numerically and not analytically as proposed for linear gradients.
  • quantization of variables involves an update of all variables.
DEVS state for TSM

Fig: The DEVS state transition graph for Time Slicing integration methods Euler and Rung Kutta 2.

DEVS state for QSS2

Fig: The DEVS state transition graph for QSS2 integration method.