`dedalus.libraries.dedalus_sphere.timesteppers`

Module Contents

STORE_LU = True

PERMC_SPEC

USE_UMFPACK

class MultistepIMEX(CoeffSystem, *args)

Base class for implicit-explicit multistep methods.

Parameters

nfields (int) – Number of fields in problem
domain (domain object) – Problem domain

Notes

These timesteppers discretize the system: M.dt(X) + L.X = F
into the general form: aj M.X(n-j) + bj L.X(n-j) = cj F(n-j)

where j runs from {0, 0, 1} to {amax, bmax, cmax}.

The system is then solved as: (a0 M + b0 L).X(n) = cj F(n-j) - aj M.X(n-j) - bj L.X(n-j)

where j runs from {1, 1, 1} to {cmax, amax, bmax}.

References

Wang and S. J. Ruuth, Journal of Computational Mathematics 26, (2008).*

Our coefficients are related to those used by Wang as:
amax = bmax = cmax = s aj = α(s-j) / k(n+s-1) bj = γ(s-j) cj = β(s-j)

step(dt, state_vector, basis, L, M, P, NL, LU): Advance solver by one timestep.

class SBDF1(CoeffSystem, *args)

1st-order semi-implicit BDF scheme [Wang 2008 eqn 2.6]

Implicit: 1st-order BDF (backward Euler) Explicit: 1st-order extrapolation (forward Euler)

amax = 1

bmax = 1

cmax = 1

classmethod compute_coefficients(timesteps, iteration)

class SBDF2(CoeffSystem, *args)

2nd-order semi-implicit BDF scheme [Wang 2008 eqn 2.8]

Implicit: 2nd-order BDF Explicit: 2nd-order extrapolation

amax = 2

bmax = 2

cmax = 2

classmethod compute_coefficients(timesteps, iteration)

class SBDF3(CoeffSystem, *args)

3rd-order semi-implicit BDF scheme [Wang 2008 eqn 2.14]

Implicit: 3rd-order BDF Explicit: 3rd-order extrapolation

amax = 3

bmax = 3

cmax = 3

classmethod compute_coefficients(timesteps, iteration)

class SBDF4(CoeffSystem, *args)

4th-order semi-implicit BDF scheme [Wang 2008 eqn 2.15]

Implicit: 4th-order BDF Explicit: 4th-order extrapolation

amax = 4

bmax = 4

cmax = 4

classmethod compute_coefficients(timesteps, iteration)

class CNAB1(CoeffSystem, *args)

1st-order Crank-Nicolson Adams-Bashforth scheme [Wang 2008 eqn 2.5.3]

Implicit: 2nd-order Crank-Nicolson Explicit: 1st-order Adams-Bashforth (forward Euler)

amax = 1

bmax = 1

cmax = 1

classmethod compute_coefficients(timesteps, iteration)

class CNAB2(CoeffSystem, *args)

2nd-order Crank-Nicolson Adams-Bashforth scheme [Wang 2008 eqn 2.9]

Implicit: 2nd-order Crank-Nicolson Explicit: 2nd-order Adams-Bashforth

amax = 2

bmax = 2

cmax = 2

classmethod compute_coefficients(timesteps, iteration)

class RungeKuttaIMEX(CoeffSystem, *args)

Base class for implicit-explicit multistep methods.

Parameters

nfields (int) – Number of fields in problem
domain (domain object) – Problem domain

Notes

These timesteppers discretize the system: M.dt(X) + L.X = F
by constructing s stages: M.X(n,i) - M.X(n,0) + k Hij L.X(n,j) = k Aij F(n,j)
where j runs from {0, 0} to {i, i-1}, and F(n,i) is evaluated at time: t(n,i) = t(n,0) + k ci
The s stages are solved as: (M + k Hii L).X(n,i) = M.X(n,0) + k Aij F(n,j) - k Hij L.X(n,j)

where j runs from {0, 0} to {i-1, i-1}.