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Complete autodocs for important functions in gauss_legendre.jl
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@mrhardman
mrhardman committed Oct 25, 2024
1 parent c8e7c9b commit 884b1c7
Showing 1 changed file with 123 additions and 11 deletions.
134 changes: 123 additions & 11 deletions moment_kinetics/src/gauss_legendre.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -1208,6 +1208,17 @@ function get_SS_local!(QQ,ielement,
return nothing
end

"""
If called for `coord.name = vperp` elemental matrix `KK` on the \$i^{th}\$ element is
```math
K_{jk} = -\\int^{v_\\perp^U}_{v_\\perp^L} \\left(v_\\perp \\frac{\\partial\\varphi_j(v_\\perp)}{\\partial v_\\perp} + \\varphi_j(v_\\perp) \\right)
\\frac{\\partial\\varphi_k(v_\\perp)}{\\partial v_\\perp} d v_\\perp
= -\\int^1_{-1} ((c_i + x s_i)l_j^\\prime(x) + l_j(x))l_k^\\prime(x) d x /s_i
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `KK` is the same as `LL` (see `get_LL_local!()).
If `explicit_BC_terms = true`, boundary terms arising from integration by parts are included at the extreme boundary points.
"""
function get_KK_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand Down Expand Up @@ -1252,8 +1263,15 @@ function get_KK_local!(QQ,ielement,
return nothing
end

# second derivative matrix with vperp^2 Jacobian factor if
# coord is vperp. Not useful for the vpa coordinate
"""
If called for `coord.name = vperp` elemental matrix `KJ` on the \$i^{th}\$ element is
```math
(KJ)_{jk} = -\\int^{v_\\perp^U}_{v_\\perp^L} \\frac{\\partial\\varphi_j(v_\\perp)}{\\partial v_\\perp}\\frac{\\partial\\varphi_k(v_\\perp)}{\\partial v_\\perp} v_\\perp^2 d v_\\perp
= -\\int^1_{-1} (c_i + x s_i)^2l_j^\\prime(x)l_k^\\prime(x) d x /s_i
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `KJ` is the same as `LL` (see `get_LL_local()!`).
"""
function get_KJ_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand All @@ -1278,6 +1296,20 @@ function get_KJ_local!(QQ,ielement,
return nothing
end

"""
If called for `coord.name = vperp` elemental matrix `LL` on the \$i^{th}\$ element is
```math
L_{jk} = -\\int^{v_\\perp^U}_{v_\\perp^L} \\frac{\\partial\\varphi_j(v_\\perp)}{\\partial v_\\perp}\\frac{\\partial\\varphi_k(v_\\perp)}{\\partial v_\\perp} v_\\perp d v_\\perp
= -\\int^1_{-1} (c_i + x s_i)l_j^\\prime(x)l_k^\\prime(x) d x /s_i
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `LL` is
```math
L_{jk} = -\\int^{v_\\|^U}_{v_\\|^L} \\frac{\\partial\\varphi_j(v_\\|)}{\\partial v_\\|}\\frac{\\partial\\varphi_k(v_\\|)}{\\partial v_\\|} d v_\\| =
-\\int^1_{-1} l_j^\\prime(x)l_k^\\prime(x) d x /s_i.
```
If `explicit_BC_terms = true`, boundary terms arising from integration by parts are included at the extreme boundary points.
"""
function get_LL_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand Down Expand Up @@ -1334,8 +1366,14 @@ function get_DD_local!(QQ, ielement, lobatto::gausslegendre_base_info,
return nothing
end

# mass matrix without vperp factor (matrix N)
# only useful for the vperp coordinate
"""
If called for `coord.name = vperp` elemental matrix `MN` on the \$i^{th}\$ element is
```math
(MN)_{jk} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\varphi_k(v_\\perp) d v_\\perp = \\int^1_{-1} l_j(x)l_k(x) s_i d x
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `MN` is the same as `MM` (see `get_MM_local!()`).
"""
function get_MN_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand All @@ -1355,8 +1393,14 @@ function get_MN_local!(QQ,ielement,
return nothing
end

# mass matrix with vperp^2 factor (matrix R)
# only useful for the vperp coordinate
"""
If called for `coord.name = vperp` elemental matrix `MR` on the \$i^{th}\$ element is
```math
(MR)_{jk} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\varphi_k(v_\\perp) v_\\perp^2 d v_\\perp = \\int^1_{-1} (c_i + s_i x)^2 l_j(x)l_k(x) s_i d x
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `MR` is the same as `MM` (see `get_MM_local!()`).
"""
function get_MR_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand All @@ -1381,8 +1425,18 @@ function get_MR_local!(QQ,ielement,
return nothing
end

# derivative matrix (matrix P, no integration by parts)
# with vperp Jacobian factor if coord is vperp (matrix P)
"""
If called for `coord.name = vperp` elemental matrix `PP` on the \$i^{th}\$ element is
```math
P_{jk} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\frac{\\partial\\varphi_k(v_\\perp)}{\\partial v_\\perp} v_\\perp d v_\\perp
= \\int^1_{-1} (c_i + x s_i)l_j(x)l_k^\\prime(x) d x
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `PP` is
```math
P_{jk} = \\int^{v_\\|^U}_{v_\\|^L} \\varphi_j(v_\\|)\\frac{\\partial\\varphi_k(v_\\|)}{\\partial v_\\|} d v_\\| = \\int^1_{-1} l_j(x)l_k^\\prime(x) d x.
```
"""
function get_PP_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand All @@ -1403,9 +1457,15 @@ function get_PP_local!(QQ,ielement,
return nothing
end

# derivative matrix (matrix P, no integration by parts)
# with vperp^2 Jacobian factor if coord is vperp (matrix U)
# not useful for vpa coordinate
"""
If called for `coord.name = vperp` elemental matrix `PP` on the \$i^{th}\$ element is
```math
(PU)_{jk} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\frac{\\partial\\varphi_k(v_\\perp)}{\\partial v_\\perp} v_\\perp^2 d v_\\perp
= \\int^1_{-1} (c_i + x s_i)^2l_j(x)l_k^\\prime(x) d x
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `PU` is the same as `PP` see `get_PP_local!()`.
"""
function get_PU_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand Down Expand Up @@ -1451,6 +1511,19 @@ function get_QQ_local!(QQ::AbstractArray{mk_float,3},
return nothing
end

"""
If called for `coord.name = vperp` elemental matrix `YY0` on the \$i^{th}\$ element is
```math
(YY0)_{jkm} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\varphi_k(v_\\perp)\\varphi_m(v_\\perp) v_\\perp d v_\\perp
= \\int^1_{-1} (c_i + x s_i)l_j(x)l_k(x)l_m(x) s_i d x
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `YY0` is
```math
(YY0)_{jkm} = \\int^{v_\\|^U}_{v_\\|^L} \\varphi_j(v_\\|)\\varphi_k(v_\\|)\\varphi_m(v_\\|) d v_\\|
= \\int^1_{-1} l_j(x)l_k(x)l_m(x) s_i d x.
```
"""
function get_YY0_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand All @@ -1471,6 +1544,19 @@ function get_YY0_local!(QQ,ielement,
return nothing
end

"""
If called for `coord.name = vperp` elemental matrix `YY1` on the \$i^{th}\$ element is
```math
(YY1)_{jkm} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\varphi_k(v_\\perp)\\frac{\\partial\\varphi_m(v_\\perp)}{\\partial v_\\perp} v_\\perp d v_\\perp
= \\int^1_{-1} (c_i + x s_i)l_j(x)l_k(x)l_m^\\prime(x) d x
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `YY1` is
```math
(YY1)_{jkm} = \\int^{v_\\|^U}_{v_\\|^L} \\varphi_j(v_\\|)\\varphi_k(v_\\|)\\frac{\\partial\\varphi_m(v_\\|)}{\\partial v_\\|} d v_\\|
= \\int^1_{-1} l_j(x)l_k(x)l_m^\\prime(x) d x.
```
"""
function get_YY1_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand All @@ -1491,6 +1577,19 @@ function get_YY1_local!(QQ,ielement,
return nothing
end

"""
If called for `coord.name = vperp` elemental matrix `YY2` on the \$i^{th}\$ element is
```math
(YY2)_{jkm} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\frac{\\partial\\varphi_k(v_\\|)}{\\partial v_\\|}\\frac{\\partial\\varphi_m(v_\\perp)}{\\partial v_\\perp} v_\\perp d v_\\perp
= \\int^1_{-1} (c_i + x s_i)l_j(x)l_k^\\prime(x)l_m^\\prime(x) d x/s_i
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `YY2` is
```math
(YY2)_{jkm} = \\int^{v_\\|^U}_{v_\\|^L} \\varphi_j(v_\\|)\\frac{\\partial\\varphi_k(v_\\|)}{\\partial v_\\|}\\frac{\\partial\\varphi_m(v_\\|)}{\\partial v_\\|} d v_\\|
= \\int^1_{-1} l_j(x)l_k^\\prime(x)l_m^\\prime(x) d x /s_i.
```
"""
function get_YY2_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand All @@ -1511,6 +1610,19 @@ function get_YY2_local!(QQ,ielement,
return nothing
end

"""
If called for `coord.name = vperp` elemental matrix `YY3` on the \$i^{th}\$ element is
```math
(YY3)_{jkm} = \\int^{v_\\perp^U}_{v_\\perp^L} \\varphi_j(v_\\perp)\\frac{\\partial\\varphi_k(v_\\|)}{\\partial v_\\|}\\varphi_m(v_\\perp) v_\\perp d v_\\perp
= \\int^1_{-1} (c_i + x s_i)l_j(x)l_k^\\prime(x)l_m(x) d x
```
with \$c_i\$ and \$s_i\$ the appropriate shift and scale factors, respectively.
Otherwise, if called for any other coordinate elemental matrix `YY3` is
```math
(YY3)_{jkm} = \\int^{v_\\|^U}_{v_\\|^L} \\varphi_j(v_\\|)\\frac{\\partial\\varphi_k(v_\\|)}{\\partial v_\\|}\\varphi_m(v_\\|) d v_\\|
= \\int^1_{-1} l_j(x)l_k^\\prime(x)l_m(x) d x.
```
"""
function get_YY3_local!(QQ,ielement,
lobatto::gausslegendre_base_info,
radau::gausslegendre_base_info,
Expand Down

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