Time discretization

The spatially discretized Navier-Stokes equations form a differential-algebraic system, with an ODE for the velocity

\frac{d u}{d t} = F (u, t) - (G p + y_{G})

subject to the algebraic constraint formed by the mass equation

M u + y_{M} = 0.

In the end of the previous section, we differentiated the mass equation in time to obtain a discrete pressure Poisson equation. This equation includes the term $\frac{d y_{M}}{d t}$ , which is non-zero if an unsteady flow of mass is added to the domain (Dirichlet boundary conditions). This term ensures that the time-continuous discrete velocity field $u (t)$ stays divergence free (conserves mass). However, if we directly discretize this system in time, the mass preservation may actually not be respected. For this, we will change the definition of the pressure such that the time-discretized velocity field is divergence free at each time step and each time sub-step (to be defined in the following).

Consider the interval $[0, T]$ for some simulation time $T$ . We will divide it into $N$ sub-intervals $[t^{n}, t^{n + 1}]$ for $n = 0, \dots, N - 1$ , with $t^{0} = 0$ , $t^{N} = T$ , and increment $Δ t^{n} = t^{n + 1} - t^{n}$ . We define $u^{n} \approx u (t^{n})$ as an approximation to the exact discrete velocity field $u (t^{n})$ , with $u^{0} = u (0)$ starting from the exact initial conditions. We say that the time integration scheme (definition of $u^{n}$ ) is accurate to the order $r$ if $u^{n} = u (t^{n}) + O (Δ t^{r})$ for all $n$ .

IncompressibleNavierStokes provides a collection of explicit and implicit Runge-Kutta methods, in addition to Adams-Bashforth Crank-Nicolson and one-leg beta method time steppers.

The code is currently not adapted to time steppers from DifferentialEquations.jl, but they may be integrated in the future.

IncompressibleNavierStokes.AbstractODEMethod Type

julia

abstract type AbstractODEMethod{T}

Abstract ODE method.

Fields

IncompressibleNavierStokes.ode_method_cache Function

julia

ode_method_cache(method, setup, u, temp)

Get time stepper cache for the given ODE method.

IncompressibleNavierStokes.runge_kutta_method Function

julia

runge_kutta_method(
    A,
    b,
    c,
    r;
    T,
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}

Get Runge Kutta method. The function checks whether the method is explicit.

p_add_solve: whether to add a pressure solve step to the method.

For implicit RK methods: newton_type, maxiter, abstol, reltol.

IncompressibleNavierStokes.create_stepper Function

julia

create_stepper(method; setup, psolver, u, temp, t, n)

Create time stepper.

IncompressibleNavierStokes.timestep Function

julia

timestep(method, stepper, Δt; θ = nothing)

Perform one time step.

Non-mutating/allocating/out-of-place version.

See also timestep!.

IncompressibleNavierStokes.timestep! Function

julia

timestep!(method, stepper, Δt; θ = nothing, cache)

Perform one time step>

Mutating/non-allocating/in-place version.

See also timestep.

Adams-Bashforth Crank-Nicolson method

IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod Type

julia

struct AdamsBashforthCrankNicolsonMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}

IMEX AB-CN: Adams-Bashforth for explicit convection (parameters α₁ and α₂) and Crank-Nicolson for implicit diffusion (implicitness θ). The method is second order for θ = 1/2.

The LU decomposition of the LHS matrix is computed every time the time step changes.

Note that, in contrast to explicit methods, the pressure from previous time steps has an influence on the accuracy of the velocity.

We here require that the time step $Δ t$ is constant. This methods uses Adams-Bashforth for the convective terms and Crank-Nicolson stepping for the diffusion and body force terms. Given the velocity field $u_{0} = u (t_{0})$ at a time $t_{0}$ and its previous value $u_{- 1} = u (t_{0} - Δ t)$ at the previous time $t_{- 1} = t_{0} - Δ t$ , the predicted velocity field $u$ at the time $t = t_{0} + Δ t$ is defined by first computing a tentative velocity:

\begin{aligned} \frac{v - u_{0}}{Δ t} & = - (α_{0} C (u_{0}, t_{0}) + α_{- 1} C (u_{- 1}, t_{- 1})) \\ + θ (D u_{0} + y_{D} (t_{0})) + (1 - θ) (D v + y_{D} (t)) \\ + θ f (t_{0}) + (1 - θ) f (t) \\ - (G p_{0} + y_{G} (t_{0})), \end{aligned}

where $θ \in [0, 1]$ is the Crank-Nicolson parameter ( $θ = \frac{1}{2}$ for second order convergence), $(α_{0}, α_{- 1}) = (\frac{3}{2}, - \frac{1}{2})$ are the Adams-Bashforth coefficients, and $v$ is a tentative velocity yet to be made divergence free. We can group the terms containing $v$ on the left hand side, to obtain

\begin{aligned} (\frac{1}{Δ t} I - (1 - θ) D) v & = (\frac{1}{Δ t} I - θ D) u_{0} \\ - (α_{0} C (u_{0}, t_{0}) + α_{- 1} C (u_{- 1}, t_{- 1})) \\ + θ y_{D} (t_{0}) + (1 - θ) y_{D} (t) \\ + θ f (t_{0}) + (1 - θ) f (t) \\ - (G p_{0} + y_{G} (t_{0})) . \end{aligned}

We can compute $v$ by inverting the positive definite matrix $(\frac{1}{Δ t} I - θ D)$ for the given right hand side using a suitable linear solver. Assuming $Δ t$ is constant, we can precompute a Cholesky factorization of this matrix before starting time stepping.

We then compute the pressure difference $Δ p$ by solving

L Δ p = W \frac{M v + y_{M} (t)}{Δ t} - W M (y_{G} (t) - y_{G} (t_{0})),

after which a divergence free velocity $u$ can be enforced:

u = v - Δ t (G Δ p + y_{G} (t) - y_{G} (t_{0})) .

A first order accurate prediction of the corresponding pressure is $p = p_{0} + Δ p$ . However, since this pressure is reused in the next time step, we perform an additional pressure solve to avoid accumulating first order errors. The resulting pressure $p$ is then accurate to the same order as $u$ .

Fields

α₁
α₂
θ
p_add_solve
method_startup

One-leg beta method

IncompressibleNavierStokes.OneLegMethod Type

julia

struct OneLegMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}

Explicit one-leg β-method following symmetry-preserving discretization of turbulent flow. See Verstappen and Veldman [14] [16] for details.

We here require that the time step $Δ t$ is constant. Given the velocity $u_{0}$ and pressure $p_{0}$ at the current time $t_{0}$ and their previous values $u_{- 1}$ and $p_{- 1}$ at the time $t_{- 1} = t_{0} - Δ t$ , we start by computing the "offstep" values $v = (1 + β) v_{0} - β v_{- 1}$ and $Q = (1 + β) p_{0} - β p_{- 1}$ for some $β = \frac{1}{2}$ .

A tentative velocity field $\tilde{v}$ is then computed as follows:

\tilde{v} = \frac{1}{β + \frac{1}{2}} (2 β u_{0} - (β - \frac{1}{2}) u_{- 1} + Δ t F (v, t) - Δ t (G Q + y_{G} (t))) .

A pressure correction $Δ p$ is obtained by solving the Poisson equation

L Δ p = \frac{β + \frac{1}{2}}{Δ t} W (M \tilde{v} + y_{M} (t)) .

Finally, the divergence free velocity field is given by

u = \tilde{v} - \frac{Δ t}{β + \frac{1}{2}} G Δ p,

while the second order accurate pressure is given by

p = 2 p_{0} - p_{- 1} + \frac{4}{3} Δ p .

Fields

β
p_add_solve
method_startup

Runge-Kutta methods

IncompressibleNavierStokes.AbstractRungeKuttaMethod Type

julia

abstract type AbstractRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractODEMethod{T}

Abstract Runge Kutta method.

Fields

IncompressibleNavierStokes.ExplicitRungeKuttaMethod Type

julia

struct ExplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}

Explicit Runge Kutta method. See Sanderse [17].

Consider the velocity field $u_{0}$ at a certain time $t_{0}$ . We will now perform one time step to $t = t_{0} + Δ t$ . For explicit Runge-Kutta methods, this time step is divided into $s$ sub-steps $t_{i} = t_{0} + Δ t_{i}$ with increment $Δ t_{i} = c_{i} Δ t$ . The final substep performs the full time step $Δ t_{s} = Δ t$ such that $t_{s} = t$ .

For $i = 1, \dots, s$ , the intermediate velocity $u_{i}$ and pressure $p_{i}$ are computed as follows:

\begin{aligned} k_{i} & = F (u_{i - 1}, t_{i - 1}) - y_{G} (t_{i - 1}) \\ v_{i} & = u_{0} + Δ t \sum_{j = 1}^{i} a_{i j} k_{j} \\ L p_{i} & = W M \frac{1}{c_{i}} \sum_{j = 1}^{i} a_{i j} k_{j} + W \frac{y_{M} (t_{i}) - y_{M} (t_{0})}{Δ t_{i}} \\ = W \frac{(M v_{i} + y_{M} (t_{i})) - (M u_{0} + y_{M} (t_{0}))}{Δ t_{i}^{n}} \\ = W \frac{M v_{i} + y_{M} (t_{i})}{Δ t_{i}^{n}} \\ u_{i} & = v_{i} - Δ t_{i} G p_{i}, \end{aligned}

where $(a_{i j})_{i j}$ are the Butcher tableau coefficients of the RK-method, with the convention $c_{i} = \sum_{j = 1}^{i} a_{i j}$ .

Finally, we return $u_{s}$ . If $u_{0} = u (t_{0})$ , we get the accuracy $u_{s} = u (t) + O (Δ t^{r + 1})$ , where $r$ is the order of the RK-method. If we perform $n$ RK time steps instead of one, starting at exact initial conditions $u^{0} = u (0)$ , then $u^{n} = u (t^{n}) + O (Δ t^{r})$ for all $n \in {1, \dots, N}$ . Note that for a given $u$ , the corresponding pressure $p$ can be calculated to the same accuracy as $u$ by doing an additional pressure projection after each outer time step $Δ t$ (if we know $\frac{d y_{M}}{d t} (t)$ ), or to first order accuracy by simply returning $p_{s}$ .

Note that each of the sub-step velocities $u_{i}$ is divergence free, after projecting the tentative velocities $v_{i}$ . This is ensured due to the judiciously chosen replacement of $\frac{d y_{M}}{d t} (t_{i})$ with $(y_{M} (t_{i}) - y_{M} (t_{0})) / Δ t_{i}$ . The space-discrete divergence-freeness is thus perfectly preserved, even though the time discretization introduces other errors.

Fields

A
b
c
r
p_add_solve

IncompressibleNavierStokes.ImplicitRungeKuttaMethod Type

julia

struct ImplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}

Implicit Runge Kutta method. See Sanderse [18].

The implicit linear system is solved at each time step using Newton's method. The newton_type may be one of the following:

:no: Replace iteration matrix with I/Δt (no Jacobian)
:approximate: Build Jacobian once before iterations only
:full: Build Jacobian at each iteration

Fields

A
b
c
r
newton_type
maxiter
abstol
reltol
p_add_solve

IncompressibleNavierStokes.RKMethods Module

Set up Butcher arrays A, b, and c, as well as and SSP coefficient r. For families of methods, optional input s is the number of stages.

Original (MATLAB) by David Ketcheson, extended by Benjamin Sanderse.

Exports

IncompressibleNavierStokes.LMWray3 Type

julia

struct LMWray3{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}

Low memory Wray 3rd order scheme. Uses 3 vector fields and one scalar field.

Fields

Explicit Methods

IncompressibleNavierStokes.RKMethods.FE11 Function

julia

FE11(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

FE11 (Forward Euler).

IncompressibleNavierStokes.RKMethods.SSP22 Function

julia

SSP22(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

SSP22.

IncompressibleNavierStokes.RKMethods.SSP42 Function

julia

SSP42(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

SSP42.

IncompressibleNavierStokes.RKMethods.SSP33 Function

julia

SSP33(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

SSP33.

IncompressibleNavierStokes.RKMethods.SSP43 Function

julia

SSP43(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

SSP43.

IncompressibleNavierStokes.RKMethods.SSP104 Function

julia

SSP104(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

SSP104.

IncompressibleNavierStokes.RKMethods.rSSPs2 Function

julia

rSSPs2(
;
    ...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
rSSPs2(
    s;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}

Rational (optimal, low-storage) s-stage 2nd order SSP.

IncompressibleNavierStokes.RKMethods.rSSPs3 Function

julia

rSSPs3(
;
    ...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
rSSPs3(
    s;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}

Rational (optimal, low-storage) s^2-stage 3rd order SSP.

IncompressibleNavierStokes.RKMethods.Wray3 Function

julia

Wray3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Wray's RK3.

IncompressibleNavierStokes.RKMethods.RK56 Function

julia

RK56(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RK56.

IncompressibleNavierStokes.RKMethods.DOPRI6 Function

julia

DOPRI6(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Dormand-Price pair.

Implicit Methods

IncompressibleNavierStokes.RKMethods.BE11 Function

julia

BE11(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Backward Euler.

IncompressibleNavierStokes.RKMethods.SDIRK34 Function

julia

SDIRK34(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

3-stage, 4th order singly diagonally implicit (SSP).

IncompressibleNavierStokes.RKMethods.ISSPm2 Function

julia

ISSPm2(
;
    ...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
ISSPm2(
    s;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}

Optimal DIRK SSP schemes of order 2.

IncompressibleNavierStokes.RKMethods.ISSPs3 Function

julia

ISSPs3(
;
    ...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
ISSPs3(
    s;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}

Optimal DIRK SSP schemes of order 3.

Half explicit methods

IncompressibleNavierStokes.RKMethods.HEM3 Function

julia

HEM3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Brasey and Hairer.

IncompressibleNavierStokes.RKMethods.HEM3BS Function

julia

HEM3BS(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

HEM3BS.

IncompressibleNavierStokes.RKMethods.HEM5 Function

julia

HEM5(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Brasey and Hairer, 5 stage, 4th order.

Classical Methods

IncompressibleNavierStokes.RKMethods.GL1 Function

julia

GL1(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

GL1.

IncompressibleNavierStokes.RKMethods.GL2 Function

julia

GL2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

GL2.

IncompressibleNavierStokes.RKMethods.GL3 Function

julia

GL3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

GL3.

IncompressibleNavierStokes.RKMethods.RIA1 Function

julia

RIA1(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

This is implicit Euler.

IncompressibleNavierStokes.RKMethods.RIA2 Function

julia

RIA2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RIA2.

IncompressibleNavierStokes.RKMethods.RIA3 Function

julia

RIA3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RIA3.

IncompressibleNavierStokes.RKMethods.RIIA1 Function

julia

RIIA1(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RIIA1.

IncompressibleNavierStokes.RKMethods.RIIA2 Function

julia

RIIA2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RIIA2.

IncompressibleNavierStokes.RKMethods.RIIA3 Function

julia

RIIA3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RIIA3.

IncompressibleNavierStokes.RKMethods.LIIIA2 Function

julia

LIIIA2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

LIIIA2.

IncompressibleNavierStokes.RKMethods.LIIIA3 Function

julia

LIIIA3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

LIIIA3.

Chebyshev methods

IncompressibleNavierStokes.RKMethods.CHDIRK3 Function

julia

CHDIRK3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Chebyshev based DIRK (not algebraically stable).

IncompressibleNavierStokes.RKMethods.CHCONS3 Function

julia

CHCONS3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

CHCONS3.

IncompressibleNavierStokes.RKMethods.CHC3 Function

julia

CHC3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Chebyshev quadrature and C(3) satisfied. Note this equals Lobatto IIIA.

IncompressibleNavierStokes.RKMethods.CHC5 Function

julia

CHC5(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

CHC5.

Miscellaneous Methods

IncompressibleNavierStokes.RKMethods.Mid22 Function

julia

Mid22(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Midpoint 22 method.

IncompressibleNavierStokes.RKMethods.MTE22 Function

julia

MTE22(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Minimal truncation error 22 method (Heun).

IncompressibleNavierStokes.RKMethods.CN22 Function

julia

CN22(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Crank-Nicholson.

IncompressibleNavierStokes.RKMethods.Heun33 Function

julia

Heun33(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Heun33.

IncompressibleNavierStokes.RKMethods.RK33C2 Function

julia

RK33C2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RK3 satisfying C(2) for i=3.

IncompressibleNavierStokes.RKMethods.RK33P2 Function

julia

RK33P2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RK3 satisfying the second order condition for the pressure.

IncompressibleNavierStokes.RKMethods.RK44 Function

julia

RK44(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Classical fourth order.

IncompressibleNavierStokes.RKMethods.RK44C2 Function

julia

RK44C2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RK4 satisfying C(2) for i=3.

IncompressibleNavierStokes.RKMethods.RK44C23 Function

julia

RK44C23(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RK4 satisfying C(2) for i=3 and c2=c3.

IncompressibleNavierStokes.RKMethods.RK44P2 Function

julia

RK44P2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

RK4 satisfying the second order condition for the pressure (but not third order).

DSRK Methods

IncompressibleNavierStokes.RKMethods.DSso2 Function

julia

DSso2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

CBM's DSRKso2.

IncompressibleNavierStokes.RKMethods.DSRK2 Function

julia

DSRK2(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

CBM's DSRK2.

IncompressibleNavierStokes.RKMethods.DSRK3 Function

julia

DSRK3(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

Zennaro's DSRK3.

"Non-SSP" Methods of Wong & Spiteri

IncompressibleNavierStokes.RKMethods.NSSP21 Function

julia

NSSP21(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

NSSP21.

IncompressibleNavierStokes.RKMethods.NSSP32 Function

julia

NSSP32(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

NSSP32.

IncompressibleNavierStokes.RKMethods.NSSP33 Function

julia

NSSP33(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

NSSP33.

IncompressibleNavierStokes.RKMethods.NSSP53 Function

julia

NSSP53(
;
    kwargs...
) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

NSSP53.