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more IMEX methods #111

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ranocha Apr 10, 2026
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ranocha Apr 10, 2026
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add HT222
ranocha Apr 10, 2026
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add ARS233
ranocha Apr 10, 2026
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ranocha Apr 10, 2026
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ranocha Apr 10, 2026
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ranocha Apr 10, 2026
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215 changes: 176 additions & 39 deletions libs/Theseus/src/imex/tableau.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -7,21 +7,81 @@ struct IMEXButcher{T1 <: AbstractArray, T2 <: AbstractArray} <: RKTableau
c_im::T2
end

"""
SP111()

The symplectic Euler method, a first-order, one-stage type I IMEX method
combining an explicit and an implicit Euler method.

## References
- Sebastiano Boscarino and Giovanni Russo (2024)
*Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review.*
[DOI: 10.1007/s40324-024-00351-x](https://doi.org/10.1007/s40324-024-00351-x)
"""
struct SP111 <: RKIMEX{1} end
function RKTableau(alg::SP111, RealT)
nstage = 1
a = zeros(RealT, nstage, nstage)
b = zeros(RealT, nstage)
b[1] = 1
c = zeros(RealT, nstage)
a_im = zeros(RealT, nstage, nstage)
a_im[1, 1] = 1
b_im = zeros(RealT, nstage)
b_im[1] = 1
c_im = zeros(RealT, nstage)
c_im[1] = 1
return IMEXButcher(a, b, c, a_im, b_im, c_im)
end

"""
H222()

A second-order, two-stage type I IMEX method.
The explicit part is strong stability preserving (SSP), the implicit part
is A-stable but not L-stable.

## References
- Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025)
Implicit-explicit methods for evolutionary partial differential equations.
[DOI: 10.1137/1.9781611978209](https://doi.org/10.1137/1.9781611978209)
"""
struct H222 <: RKIMEX{2} end
function RKTableau(alg::H222, RealT)
nstage = 2
a = zeros(RealT, nstage, nstage)
a[2, 1] = 1
b = zeros(RealT, nstage)
b[1] = 1 // 2
b[2] = 1 // 2
c = zeros(RealT, nstage)
c[2] = 1
a_im = zeros(RealT, nstage, nstage)
a_im[1, 1] = 1 // 2
a_im[2, 2] = 1 // 2
b_im = zeros(RealT, nstage)
b_im[1] = 1 // 2
b_im[2] = 1 // 2
c_im = zeros(RealT, nstage)
c_im[1] = 1 // 2
c_im[2] = 1 // 2
return IMEXButcher(a, b, c, a_im, b_im, c_im)
end

"""
SSP2222()

A second-order type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
A second-order, two-stage type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
is L-stable.

## References
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
[DOI: 10.1007/s10915-004-4636-4](https://doi.org/10.1007/s10915-004-4636-4)
- Sebastiano Boscarino and Giovanni Russo (2024)
*Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review.*
- Sebastiano Boscarino and Giovanni Russo (2024)
*Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review.*
[DOI: 10.1007/s40324-024-00351-x](https://doi.org/10.1007/s40324-024-00351-x)
"""
struct SSP2222 <: RKIMEX{2} end
Expand Down Expand Up @@ -52,14 +112,14 @@ end
"""
SSP2322()

A second-order type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
A second-order, three-stage type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
is stiffly accurate (SA) and thus L-stable.

## References
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
[DOI: 10.1007/s10915-004-4636-4](https://doi.org/10.1007/s10915-004-4636-4)
"""
struct SSP2322 <: RKIMEX{3} end
Expand Down Expand Up @@ -91,17 +151,17 @@ end
"""
SSP2332()

A second-order type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
A second-order, three-stage type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
is stiffly accurate (SA) and thus L-stable.

## References
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
[DOI: 10.1007/s10915-004-4636-4](https://doi.org/10.1007/s10915-004-4636-4)
- Sebastiano Boscarino and Giovanni Russo (2024)
*Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review.*
- Sebastiano Boscarino and Giovanni Russo (2024)
*Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review.*
[DOI: 10.1007/s40324-024-00351-x](https://doi.org/10.1007/s40324-024-00351-x)
"""
struct SSP2332 <: RKIMEX{3} end
Expand Down Expand Up @@ -139,14 +199,14 @@ end
"""
SSP3332()

A third-order type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
is L-stable.
A second-order, three-stage type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP) and third-order accurate,
the implicit part is L-stable.

## References
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
[DOI: 10.1007/s10915-004-4636-4](https://doi.org/10.1007/s10915-004-4636-4)
"""
struct SSP3332 <: RKIMEX{3} end
Expand Down Expand Up @@ -185,17 +245,17 @@ end
"""
SSP3433()

A third-order type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP), the implicit part
is L-stable.
A third-order, four-stage type I IMEX method developed by Pareschi and Russo (2005).
The explicit part is strong stability preserving (SSP) and third-order accurate,
the implicit part is L-stable.

## References
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
- Lorenzo Pareschi and Giovanni Russo (2005)
*Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation.*
*Journal of Computational Physics* 203(2):469–491.
[DOI: 10.1007/s10915-004-4636-4](https://doi.org/10.1007/s10915-004-4636-4)
- Sebastiano Boscarino and Giovanni Russo (2024)
*Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review.*
- Sebastiano Boscarino and Giovanni Russo (2024)
*Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review.*
[DOI: 10.1007/s40324-024-00351-x](https://doi.org/10.1007/s40324-024-00351-x)
"""
struct SSP3433 <: RKIMEX{4} end
Expand Down Expand Up @@ -237,11 +297,44 @@ function RKTableau(alg::SSP3433, RealT)
return IMEXButcher(a, b, c, a_im, b_im, c_im)
end

"""
HT222()

A second-order, two-stage type II IMEX method.
The explicit part is strong stability preserving (SSP), the implicit part
is A-stable but not L-stable.

## References
- Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025)
Implicit-explicit methods for evolutionary partial differential equations.
[DOI: 10.1137/1.9781611978209](https://doi.org/10.1137/1.9781611978209)
"""
struct HT222 <: RKIMEX{2} end
function RKTableau(alg::HT222, RealT)
nstage = 2
a = zeros(RealT, nstage, nstage)
a[2, 1] = 1
b = zeros(RealT, nstage)
b[1] = 1 // 2
b[2] = 1 // 2
c = zeros(RealT, nstage)
c[2] = 1
a_im = zeros(RealT, nstage, nstage)
a_im[2, 1] = 1 // 2
a_im[2, 2] = 1 // 2
b_im = zeros(RealT, nstage)
b_im[1] = 1 // 2
b_im[2] = 1 // 2
c_im = zeros(RealT, nstage)
c_im[2] = 1
return IMEXButcher(a, b, c, a_im, b_im, c_im)
end

"""
ARS111()

A first-order, globally stiffly accurate (GSA) type II IMEX method developed by
Ascher, Ruuth, and Spiteri (1997).
A first-order, effectively one-stage, globally stiffly accurate (GSA) type II IMEX method
developed by Ascher, Ruuth, and Spiteri (1997).

## References

Expand Down Expand Up @@ -279,8 +372,8 @@ end
"""
ARS222()

A second-order, globally stiffly accurate (GSA) type II IMEX method developed by
Ascher, Ruuth, and Spiteri (1997).
A second-order, effectively two-stage, globally stiffly accurate (GSA) type II IMEX method
developed by Ascher, Ruuth, and Spiteri (1997).

## References

Expand Down Expand Up @@ -325,11 +418,55 @@ function RKTableau(alg::ARS222, RealT)
return IMEXButcher(a, b, c, a_im, b_im, c_im)
end

"""
ARS233()

A third-order, effectively three-stage type II IMEX method
developed by Ascher, Ruuth, and Spiteri (1997).
The implicit part is A-stable but not L-stable.

## References

- Uri M. Ascher, Steven J. Ruuth, and Raymond J Spiteri (1997)
Implicit-explicit Runge-Kutta methods for time-dependent
partial differential equations.
[DOI: 10.1016/S0168-9274(97)00056-1](https://doi.org/10.1016/S0168-9274(97)00056-1)
- Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025)
Implicit-explicit methods for evolutionary partial differential equations.
[DOI: 10.1137/1.9781611978209](https://doi.org/10.1137/1.9781611978209)
"""
struct ARS233 <: RKIMEX{3} end
function RKTableau(alg::ARS233, RealT)
nstage = 3
gamma = (3 + sqrt(3)) / 6
a = zeros(RealT, nstage, nstage)
a[2, 1] = gamma
a[3, 1] = gamma - 1
a[3, 2] = 2 * (1 - gamma)
b = zeros(RealT, nstage)
b[2] = 1 // 2
b[3] = 1 // 2
c = zeros(RealT, nstage)
c[2] = gamma
c[3] = 1 - gamma
a_im = zeros(RealT, nstage, nstage)
a_im[2, 2] = gamma
a_im[3, 2] = 1 - 2 * gamma
a_im[3, 3] = gamma
b_im = zeros(RealT, nstage)
b_im[2] = 1 // 2
b_im[3] = 1 // 2
c_im = zeros(RealT, nstage)
c_im[2] = gamma
c_im[3] = 1 - gamma
return IMEXButcher(a, b, c, a_im, b_im, c_im)
end

"""
ARS443()

A third-order, globally stiffly accurate (GSA) type II IMEX method developed by
Ascher, Ruuth, and Spiteri (1997).
A third-order, effectively four-stage, globally stiffly accurate (GSA) type II IMEX method
developed by Ascher, Ruuth, and Spiteri (1997).

## References

Expand Down
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