TimeIntegrators

The time integration module implemented in Mantis is based on the framework developed by (Vos et al., 2011). This framework allows for an easy implementation of a variety of explicit, implicit, and implicit-explicit (IMEX) time stepping schemes, and is based on the concept of general linear methods (GLMs). See, for example, (Butcher, 2006), for more details. These methods are applicable to both ODEs and PDEs, so that both are available in Mantis.

GLMs: Notation and Theory

General linear methods can be characterised as follows (see (Butcher, 2006), (Vos et al., 2011)). Consider the initial value problem defined as the ODE

\frac{d y}{d t} = f (y), y (t_{0}) = y_{0},

where $f : R^{N} \to R^{N}$ . The $n$ -th (time) step of the GLM comprised of $r$ (integrator) steps and $s$ stages is then formulated as

\begin{aligned} Y_{i} & = Δ t \sum_{j = 1}^{s} a_{i j} F_{j} + \sum_{j = 1}^{r} u_{i j} y_{j}^{n - 1}, i = 1, \dots, s, \\ y_{i}^{n} & = Δ t \sum_{j = 1}^{s} b_{i j} F_{j} + \sum_{j = 1}^{r} v_{i j} y_{j}^{n - 1}, i = 1, \dots, r, \end{aligned}

where $Y_{i}$ are called the stage values and $F_{i}$ are called the stage derivatives. These two quantities are related by the differential equation

F_{i} = f (Y_{i}) .

The above formulation can be cast into the following matrix form

[\begin{matrix} Y \\ y^{n} \end{matrix}] = [\begin{matrix} A \otimes I & U \otimes I \\ B \otimes I & V \otimes I \end{matrix}] [\begin{matrix} Δ t Y \\ y^{n - 1} \end{matrix}],

which is often simplified (with some abuse of notation) to

[\begin{matrix} Y \\ y^{n} \end{matrix}] = [\begin{matrix} A & U \\ B & V \end{matrix}] [\begin{matrix} Δ t Y \\ y^{n - 1} \end{matrix}] .

Either way, the vectors $y$ (in/output approximations), $Y$ (stage values), and $F$ (stage derivatives) are defined as

y^{n - 1} = [\begin{matrix} y_{1}^{n - 1} \\ y_{2}^{n - 1} \\ ⋮ \\ y_{r}^{n - 1} \end{matrix}], y^{n} = [\begin{matrix} y_{1}^{n} \\ y_{2}^{n} \\ ⋮ \\ y_{r}^{n} \end{matrix}], Y = [\begin{matrix} Y_{1} \\ Y_{2} \\ ⋮ \\ Y_{s} \end{matrix}], F = [\begin{matrix} F_{1} \\ F_{2} \\ ⋮ \\ F_{s} \end{matrix}] .

It is important to note that the in/output vectors can contain more than just the solution. The exact content depends on the specific method, but often includes previously computed stage derivatives. This is particularly important when creating and/or initialising new methods.

GLMs: Characterising a GLM

Any time integrator that fits in the above framework can thus be characterised by the four matrices $A$ , $B$ , $U$ , $V$ , and the layout of the in- and output vectors $y$ . In addition, every time integrator will also need a vector $C$ , which keeps track of the time at which the stages are evaluated. In Mantis, $A$ , $B$ , $U$ , $V$ , and $C$ are stored in the time integrator structs (see this section below). The matrices have the following sizes:

Matrix	Size
$A$	`num_stages` x `num_stages`
$B$	`num_steps` x `num_stages`
$U$	`num_stages` x `num_steps`
$V$	`num_steps` x `num_steps`
$C$	`num_stages` x `1`

GLMs: Extension to IMEX Schemes

The framework introduced by (Vos et al., 2011) extends the GLM idea to IMEX integrators. The ODE from the previous section is now split into

\frac{d y}{d t} = f (y) + g (y), y (t_{0}) = y_{0},

where $f : R^{N} \to R^{N}$ and $g : R^{N} \to R^{N}$ . The $f$ -part represent the part of the ODE that is treated explicitly, while the $g$ -part is treated implicitly. The $n$ -th (time) step of the IMEX-GLM comprised of $r$ (integrator) steps and $s$ stages is then formulated as

\begin{aligned} Y_{i} & = Δ t \sum_{j = 1}^{s} a_{i j}^{I M} G_{j} + Δ t \sum_{j = 1}^{s} a_{i j}^{E X} F_{j} + \sum_{j = 1}^{r} u_{i j} y_{j}^{n - 1}, i = 1, \dots, s, \\ y_{i}^{n} & = Δ t \sum_{j = 1}^{s} b_{i j}^{I M} G_{j} + Δ t \sum_{j = 1}^{s} b_{i j}^{E X} F_{j} + \sum_{j = 1}^{r} v_{i j} y_{j}^{n - 1}, i = 1, \dots, r, \end{aligned}

where $Y_{i}$ are called the stage values and $F_{i}$ and $G_{i}$ are called the (explicit and implicit) stage derivatives. These quantities are related by the differential equation

F_{i} = f (Y_{i}), G_{i} = g (Y_{i}) .

The matrix form is obtained in the same way as in the previous section.

IMEX GLMs are characterised in the same way as described in GLMs: Characterising a GLM. The only difference is that an IMEX GLM will have two $A$ and $B$ matrices, and two $C$ vectors : one for the explicit part and one for the implicit part.

What are time integrators in `Mantis`?

The top-level type within the TimeIntegrators module is the AbstractTimeIntegrator{num_stages, num_steps} type.

Mantis.TimeIntegrators.AbstractTimeIntegrator Type

julia

AbstractTimeIntegrator{num_stages, num_steps}

Supertype for all time integrators.

Type parameters

num_stages: The number of stages for a multi-step scheme (such as the Runge-Kutta family). Since every scheme is at least a single-stage scheme, num_stages >= 1.
num_steps: The number of steps for a multi-step scheme (such as the Adams-Bashforth family). Since every scheme is at least a single-step scheme, num_steps >= 1.

TimeIntegrators ​

GLMs: Notation and Theory ​

GLMs: Characterising a GLM ​

GLMs: Extension to IMEX Schemes ​

What are time integrators in Mantis? ​

The solution objects ​

Adding problem-specific information ​

Time Integration ​

Time Integrate ​

Initialisation ​

Pre-implemented schemes ​

Adding your own scheme ​

TimeIntegrators

GLMs: Notation and Theory

GLMs: Characterising a GLM

GLMs: Extension to IMEX Schemes

What are time integrators in `Mantis`?

The solution objects

Adding problem-specific information

Time Integration

Time Integrate

Initialisation

Pre-implemented schemes

Adding your own scheme