adam_flume_cpu_dissipation_kernels

ADAM, FLUME CPU kernels of the dissipative face fluxes (issue #65, M4).

The viscous and heat-conduction face fluxes are ADDED to the inviscid face fluxes of direction d (fl), before the seam accumulation and the flux difference: the AMR reflux and the inter-realm register then see the total flux. They read W = (u, v, w, T) from the auxiliary variables, whose indexes are those of every model, so one kernel serves Euler and MHD; the host selects the kernel of the order (2 or 4, [numerics] dissipative_order), never a branch inside the loops.

Face (i,j,k) lies between cells (i,j,k) and (i,j,k) + e_d; m counts the cells along d from the left one (m = 0). A null direction has a zero step, so its derivatives are differences of a cell with itself, zero exactly, and its ghost cells are never read.

Order 2: h = G(W_f, grad W_f), W_f = (W_0 + W_1) / 2, normal derivative (W_1 - W_0) / dx, tangential ones the mean of the two cells' central differences.

Order 4 (Shu & Osher 1989, J. Comput. Phys. 83, 32): the conservative flux of a point-value scheme is h = G_f - dx^2 / 24 G''_f + O(dx^4), with

 W_f       = (9 (W_0 + W_1) - (W_-1 + W_2)) / 16
 dW/dn_f   = (27 (W_1 - W_0) - (W_2 - W_-1)) / (24 dx)
 dW/dt_f   = (9 (D_0 + D_1) - (D_-1 + D_2)) / 16,  D_m = (8 (W_+1 - W_-1) - (W_+2 - W_-2)) / (12 dt)  at cell m
 G''_f dx^2 = (G_2 - G_1 - G_0 + G_-1) / 2,        G_m from 2nd-order central gradients at cell m

so h = G_f - (G_2 - G_1 - G_0 + G_-1) / 48. Across a strong temperature jump the 4-point face temperature can undershoot below zero (a power law is then NaN): where it is not positive, the face takes the mean of its two cells. Cells -2 ... 3 along d and -2 ... 2 along the tangents are read: ngc >= 3. The correction is built from 2nd-order cell fluxes for every term, linear or not, so for constant mu the scheme is a 4th-order u_xx on 6 points, not the compact 5-point one; the order is verified by measurement.

The Ohmic fluxes (MHD, issue #65 P3) use the same stencils on W = B (auxiliary IA_BX..IA_BZ), adding to the field and energy fluxes: the kernels add_resistive_fluxes_o2/o4, called by the host only when the resistivity is on.

Source: src/app/flume/cpu/adam_flume_cpu_dissipation_kernels.F90

Dependencies

Contents

• add_dissipative_fluxes_o2
• add_dissipative_fluxes_o4
• add_sgs_fluxes_o2
• add_sgs_fluxes_o4
• compute_sgs_dissipation
• compute_lambda_sgs
• compute_turbulence_integrals
• add_resistive_fluxes_o2
• add_resistive_fluxes_o4
• steps
• inverse_steps

Variables

Name	Type	Attributes	Description
IW	integer(kind=I4P)	parameter	Auxiliary indexes of W = (u, v, w, T).
IB	integer(kind=I4P)	parameter	Auxiliary indexes of the magnetic field.

Subroutines

add_dissipative_fluxes_o2

Add the 2nd-order dissipative face fluxes of direction d to fl.

subroutine add_dissipative_fluxes_o2(d, di, dj, dk, ni, nj, nk, ngc, blocks_number, mu0, k0, tref, suth, omega_mu, omega_k, dxyz, is_null, q_aux, fl)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
di	integer(kind=I4P)	in		Unit step along d.
dj	integer(kind=I4P)	in		Unit step along d.
dk	integer(kind=I4P)	in		Unit step along d.
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
mu0	real(kind=R8P)	in		Viscosity and conductivity.
k0	real(kind=R8P)	in		Viscosity and conductivity.
tref	real(kind=R8P)	in		Temperature laws.
suth	real(kind=R8P)	in		Temperature laws.
omega_mu	real(kind=R8P)	in		Temperature laws.
omega_k	real(kind=R8P)	in		Temperature laws.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
fl	real(kind=R8P)	inout		Face fluxes of direction d.

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add_dissipative_fluxes_o4

Add the 4th-order conservative dissipative face fluxes of direction d to fl.

subroutine add_dissipative_fluxes_o4(d, di, dj, dk, ni, nj, nk, ngc, blocks_number, mu0, k0, tref, suth, omega_mu, omega_k, dxyz, is_null, q_aux, fl)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
di	integer(kind=I4P)	in		Unit step along d.
dj	integer(kind=I4P)	in		Unit step along d.
dk	integer(kind=I4P)	in		Unit step along d.
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
mu0	real(kind=R8P)	in		Viscosity and conductivity.
k0	real(kind=R8P)	in		Viscosity and conductivity.
tref	real(kind=R8P)	in		Temperature laws.
suth	real(kind=R8P)	in		Temperature laws.
omega_mu	real(kind=R8P)	in		Temperature laws.
omega_k	real(kind=R8P)	in		Temperature laws.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
fl	real(kind=R8P)	inout		Face fluxes of direction d.

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add_sgs_fluxes_o2

Add to fl the subgrid-scale fluxes of direction d (2nd order), those of add_dissipative_fluxes_o2 with mu = mu_t and k = mu_t cp / Pr_t in place of the laws (issue #119, T1): the mean of the two cells at the face, whose stencil stays positive, the cell value in the correction.

subroutine add_sgs_fluxes_o2(d, di, dj, dk, ni, nj, nk, ngc, blocks_number, cp_prt, dxyz, is_null, q_aux, mu_t, fl)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
di	integer(kind=I4P)	in		Unit step along d.
dj	integer(kind=I4P)	in		Unit step along d.
dk	integer(kind=I4P)	in		Unit step along d.
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
cp_prt	real(kind=R8P)	in		cp / Pr_t.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
mu_t	real(kind=R8P)	in		Eddy viscosity.
fl	real(kind=R8P)	inout		Face fluxes of direction d.

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add_sgs_fluxes_o4

Add to fl the subgrid-scale fluxes of direction d (4th order), those of add_dissipative_fluxes_o4 with mu = mu_t and k = mu_t cp / Pr_t in place of the laws (issue #119, T1): the mean of the two cells at the face, whose stencil stays positive, the cell value in the correction.

subroutine add_sgs_fluxes_o4(d, di, dj, dk, ni, nj, nk, ngc, blocks_number, cp_prt, dxyz, is_null, q_aux, mu_t, fl)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
di	integer(kind=I4P)	in		Unit step along d.
dj	integer(kind=I4P)	in		Unit step along d.
dk	integer(kind=I4P)	in		Unit step along d.
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
cp_prt	real(kind=R8P)	in		cp / Pr_t.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
mu_t	real(kind=R8P)	in		Eddy viscosity.
fl	real(kind=R8P)	inout		Face fluxes of direction d.

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compute_sgs_dissipation

Compute the volume integral of the subgrid-scale dissipation tau_sgs_ij d u_i / d x_j on this rank (issue #119, T1), as eps of compute_turbulence_integrals with mu = mu_t; the column eps_sgs of the turbulence history.

subroutine compute_sgs_dissipation(ni, nj, nk, ngc, blocks_number, dxyz, is_null, q_aux, mu_t, eps)

Arguments

Name	Type	Intent	Attributes	Description
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
mu_t	real(kind=R8P)	in		Eddy viscosity.
eps	real(kind=R8P)	out		The integral.

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compute_lambda_sgs

Compute max(2 factor mu_t / rho sum_d 1 / dx_d^2) over the interior cells (issue #119, T1): the part of the time step limit of the eddy viscosity, factor = max(4/3, gamma / Pr_t).

subroutine compute_lambda_sgs(ni, nj, nk, ngc, blocks_number, factor, dxyz, is_null, q_aux, mu_t, lambda_sgs)

Arguments

Name	Type	Intent	Attributes	Description
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
factor	real(kind=R8P)	in		max(4/3, gamma / Pr_t).
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
mu_t	real(kind=R8P)	in		Eddy viscosity.
lambda_sgs	real(kind=R8P)	out		The maximum.

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compute_turbulence_integrals

Compute the volume integrals of the turbulence history over the interior cells of this rank (issue #119, T0.1), from 4th-order central velocity gradients; the volume includes the null directions, whose gradients are zero (issue #37).

subroutine compute_turbulence_integrals(ni, nj, nk, ngc, blocks_number, mu0, tref, suth, omega_mu, dxyz, is_null, q_aux, integrals)

Arguments

Name	Type	Intent	Attributes	Description
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
mu0	real(kind=R8P)	in		Viscosity law.
tref	real(kind=R8P)	in		Viscosity law.
suth	real(kind=R8P)	in		Viscosity law.
omega_mu	real(kind=R8P)	in		Viscosity law.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
integrals	real(kind=R8P)	out		Integrals (compute_turbulence_integrands).

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add_resistive_fluxes_o2

Add the 2nd-order Ohmic face fluxes of direction d to fl (MHD).

subroutine add_resistive_fluxes_o2(d, di, dj, dk, ni, nj, nk, ngc, blocks_number, eta, dxyz, is_null, q_aux, fl)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
di	integer(kind=I4P)	in		Unit step along d.
dj	integer(kind=I4P)	in		Unit step along d.
dk	integer(kind=I4P)	in		Unit step along d.
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
eta	real(kind=R8P)	in		Magnetic diffusivity.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
fl	real(kind=R8P)	inout		Face fluxes of direction d.

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add_resistive_fluxes_o4

Add the 4th-order conservative Ohmic face fluxes of direction d to fl (MHD): the stencils of add_dissipative_fluxes_o4 on W = B, with the Shu-Osher correction.

subroutine add_resistive_fluxes_o4(d, di, dj, dk, ni, nj, nk, ngc, blocks_number, eta, dxyz, is_null, q_aux, fl)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
di	integer(kind=I4P)	in		Unit step along d.
dj	integer(kind=I4P)	in		Unit step along d.
dk	integer(kind=I4P)	in		Unit step along d.
ni	integer(kind=I4P)	in		Grid dimensions.
nj	integer(kind=I4P)	in		Grid dimensions.
nk	integer(kind=I4P)	in		Grid dimensions.
ngc	integer(kind=I4P)	in		Grid dimensions.
blocks_number	integer(kind=I4P)	in		Actual blocks number.
eta	real(kind=R8P)	in		Magnetic diffusivity.
dxyz	real(kind=R8P)	in		Blocks space steps [3, nb].
is_null	logical	in		Null directions.
q_aux	real(kind=R8P)	in		Auxiliary variables.
fl	real(kind=R8P)	inout		Face fluxes of direction d.

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Functions

steps

Return the unit steps along each direction, o(:,t) = e_t, zero along the null directions.

Attributes: pure

Returns: integer(kind=I4P)

function steps(is_null) result(o)

Arguments

Name	Type	Intent	Attributes	Description
is_null	logical	in		Null directions.

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inverse_steps

Return the inverse space steps, zero along the null directions.

Attributes: pure

Returns: real(kind=R8P)

function inverse_steps(ds, is_null) result(rdx)

Arguments

Name	Type	Intent	Attributes	Description
ds	real(kind=R8P)	in		Space steps.
is_null	logical	in		Null directions.

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