Mathematical background

If you the state of a process today and you know the physical laws that describes how this process changes, then you can formulate a differential equation to compute the state that that process a later time. For example if you know the whether today and the equations for modeling the change in the whether then we can predict the whether tomorrow.

Ordinary differential equations (ODEs) are equations of the form

\[\frac{d \mathbf{x}}{d t} = f(\mathbf{x}, t; \mathbf{p})\]

Here \(\mathbf{x}\) is a vector of state variables, \(t\) is the time, \(p\) are some known parameters, and \(f\) describes how \(\mathbf{x}\) evolves over time.

A simple example

A very simple ODE is the model of an exponential growth

\[\frac{{\rm d}u}{{\rm d}t} = a u\]

where \(a\) is the growth constant. Here \(u\) could for example be a population of some species that grows exponentially with a factor \(a\), i.e at any point in time the change in \(u\) is proportionally \(u\) times some growth factor.

One way to solve this equations is to use the definition of the derivative

\[\frac{{\rm d}u}{{\rm d}t} = \mathrm{lim}_{h \mapsto 0 }\frac{u(t + h) - u(t)}{h}\]

to find an approximate solution

\[\frac{{\rm d}u}{{\rm d}t} \approx \frac{u(t + h) - u(t)}{h}\]

for a sufficiently small \(h\). We can then instead write

\[\begin{split}\frac{{\rm d}u}{{\rm d}t} = \approx \frac{u(t + h) - u(t)}{h} = a u(t) \\ \implies u(t + h) = u(t) + h a u(t)\end{split}\]

If we know the solution at time \(t\) we can now approximate the solution at time \(t + h\). This method is known as the explicit Euler method of forward Euler. If we instead had evaluated the right hand side at \(t+u\), i.e

\[\begin{split}\frac{{\rm d}u}{{\rm d}t} = \approx \frac{u(t + h) - u(t)}{h} = a u(t + h) \\ \implies u(t + h) = \frac{u(t)}{1 - ha}\end{split}\]

we would obtain a different scheme known as the implicit euler method or backward Euler.

Different schemes are useful in different situations. For examples for stiff ODEs, you might need to use an implicit scheme rather than an explicit scheme. In other cases it might be beneficial to apply adaptive time stepping to speed up the solving time.

Schemes in goss

In goss we have implemented 11 different schemes which could

Name	Explicit/Implicit	Adaptive/Nonadaptive
ExplicitEuler	Explicit	Nonadaptive
RK2	Explicit	Nonadaptive
RK4	Explicit	Nonadaptive
RL1	Explicit	Nonadaptive
RL2	Explicit	Nonadaptive
GRL1	Explicit	Nonadaptive
GRL2	Explicit	Nonadaptive
ImplicitEuler	Implicit	Nonadaptive
ThetaSolver	Implicit	Nonadaptive
RKF32	Explicit	Adaptive
ESDIRK23a	Implicit	Adaptive