MOIRA is a Python software that generates the modified equation for a time dependent partial differential equation (PDE). Given certain finite difference scheme for the PDE, MOIRA will obtain the amplification factor and return the latex form of the modified equation.
To use MOIRA, one have to instantiate a first order time dependent differential equation object by calling its constructor that has the following signature
DifferentialEquation(dependentVar,independentVars,indices=[i, j, k], timeIndex=n)Once the user constructed an object of type DifferentialEquation, the next step is to start constructing the right hand side for this object. Two methods are available to achieve this: the first is to use the function member expr which has the following signature
expr(points, direction, order, time)The second method is to use the dependent variable name, the indices, and the differential elements of the independent variable defined by the user, d<independentVars>. The signature of the member function <dependentVars> is as follow
<dependentVar>(time, **kwargs)time here is the discrete time at which the expression of the dependent variable is evaluated ex: n+1, n, ... . kwargsare indicies of the spatial points at which this expression is evaluated, ex: x=i+1, y=j, ....
After the construction of the rhs expression using one the two previous methods or a combination of both the user can set the rhs for this differential equation by calling the member function set_rhs that have the following signature
set_rhs(expression)where expression is a symbolic expression of the rhs constructed using the previously described methods.
Now the member function, modified_equation(...), can be used to generate the modified equation up to certain number of terms that the user specify. The signature of modified_equation is as follow
modified_equation(nterms)where nterms is a positive integer that indicates the total number of terms in the modified equation.
After calling the member function modified_equation(...), one can call the latex() member function that returns the latex representation of the modified equation.
Below are some examples of using the MOIRA software.
Starting with an example of using expr(...) with the advection equation in one dimension using Forward Euler for time discretization and UPWIND for spatial discretization
from src.MOIRA import DifferentialEquation, i, n
# defining the advection velocity
a= symbols('a')
#constructing a time dependent differential equation
DE = DifferentialEquation(dependentVar='u',independentVars=['x'])
# method I of constructing the rhs:
advectionTerm1 = DE.expr(points=[-1, 0], direction='x', order=1, time=n)
# setting the rhs of the differential equation
DE.set_rhs(- a * advectionTerm1 )
# computing the modified equation up to two terms
DE.modified_equation(nterms=2)
# displaying the modified equation in latex form
pretty_print(DE.latex())Similarly, one can use the <dependentVar>(...) instead of expr(...) to construct the discretization of the rhs
from src.MOIRA import DifferentialEquation, i, n
# defining the advection velocity
a= symbols('a')
#constructing a time dependent differential equation
DE = DifferentialEquation(dependentVar='u',independentVars=['x'])
# method II of constructing the rhs:
advectionTerm = (DE.u(time=n, x=i) - DE.u(time=n, x=i-1))/DE.dx
# setting the rhs of the differential equation
DE.set_rhs(- a * advectionTerm )
# computing the modified equation up to two terms
DE.modified_equation(nterms=2)
# displaying the modified equation in latex form
pretty_print(DE.latex())