adam_flume_euler_baro_library

ADAM, FLUME pointwise barotropic Euler physics shared by the CPU and FNL backends (issue #100, M13 P3).

The body of adam_flume_euler_iso_library on a piecewise polytrope, [eos] type = barotropic: the energy slot is inert (D-M13-1) and the EOS argument named gamma carries the id of the realm's table in adam_flume_eos_tables_library (as a real). Only the four EOS hooks below differ from the isothermal library; the eigensystem, the Roe average and the LLF/HLL/HLLC fluxes are the body's, with c = sqrt(dP/drho) of each state.

ADAM, FLUME pointwise barotropic Euler physics, the body of adam_flume_euler_iso_library and adam_flume_euler_baro_library (issue #100, M13).

The including module renames NV_K, NV_AUX_K (its widths) and NV_BASE = NV_EULER, NV_AUX_BASE = NV_AUX (the Euler widths): the components NV_BASE+1 ... NV_K are advected scalars, as in the Euler body; with NV_K = NV_BASE every scalar loop is empty. The public interface is the Euler one, routine by routine, so the model-agnostic kernel bodies call either library unchanged; the EOS parameter keeps the Euler keyword gamma, and HERE IT IS physics%eos_arg: the isothermal sound speed squared c0^2, or the id of a barotropic table (adam_flume_eos_tables_library).

A barotropic gas, P = P(rho) (D-M13-1): the energy slot IQ_RE is inert. Its flux is zero, every Riemann flux forces it to zero, its eigenvalue is zero and its eigenvector the unit vector, so it keeps its initial value, 0. The including module provides the EOS as four pure hooks, the only places it enters: eos_pressure(gamma, rho), eos_temperature(gamma, R, rho) (P / (rho R)), eos_sound_speed(gamma, rho) (sqrt(dP/drho)) and eos_roe_sound_speed(gamma, rhoL, rhoR, pL, pR), the speed of the Roe average. The total enthalpy is undefined (0).

Eigensystem in direction d: normal n = e_d, tangents taken cyclically t1 = e_{mod(d,3)+1}, t2 = e_{mod(d+1,3)+1}, waves in the Euler order with the inert slot in the place of the entropy wave:

 r1 = (1, u - a n, 0)       l1 = ((u_n + a) / (2 a), -n / (2 a), 0)       lambda1 = u_n - a
 r2 = (0, 0, 1)             l2 = (0, 0, 1)                               lambda2 = 0      (the inert slot)
 r3 = (0, t1, 0)            l3 = (-u.t1, t1, 0)                          lambda3 = u_n
 r4 = (0, t2, 0)            l4 = (-u.t2, t2, 0)                          lambda4 = u_n
 r5 = (1, u + a n, 0)       l5 = (-(u_n - a) / (2 a), n / (2 a), 0)      lambda5 = u_n + a

checked against dF/dU in src/tests/flume/verification/eos/eos_oracle.py (eigen). The Roe average of two barotropic states is the sqrt(rho)-weighted velocity with the speed of eos_roe_sound_speed: c0 for the isothermal gas, the secant sqrt([P] / [rho]) for a table (the Roe property [F] = A~ [U] needs exactly that speed).

Source: src/app/flume/common/adam_flume_euler_baro_library.F90

Dependencies

Contents

• compute_eigenvalues
• compute_eigenvectors
• compute_face_flux_back_projection
• compute_face_interpolation_fields
• compute_face_split_fluxes
• compute_face_states
• compute_flux
• compute_roe_average
• compute_riemann_hll
• compute_riemann_hllc
• compute_riemann_llf
• conservative_to_auxiliary
• primitive_to_conservative
• hll_flux
• compute_state_magnitudes
• eos_pressure
• eos_temperature
• eos_sound_speed
• eos_roe_sound_speed

Variables

Name	Type	Attributes	Description
ROE_SECANT_MIN	real(kind=R8P)	parameter	Relative density gap below which the Roe speed is c(rho mean).

Subroutines

compute_eigenvalues

Compute the eigenvalues of the flux Jacobian in direction d: (u_n - a, 0, u_n, u_n, u_n + a).

Attributes: pure

subroutine compute_eigenvalues(d, qa, lambda)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
qa	real(kind=R8P)	in		Auxiliary variables.
lambda	real(kind=R8P)	out		Eigenvalues.

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compute_eigenvectors

Compute the left (rows) and right (columns) eigenvectors of the flux Jacobian in direction d at state qa.

Attributes: pure

subroutine compute_eigenvectors(gamma, d, qa, el, er)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2 (unused: a is in qa).
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
qa	real(kind=R8P)	in		Auxiliary variables (e.g. a Roe average).
el	real(kind=R8P)	out		Left eigenvectors, el(k,:) = l_k.
er	real(kind=R8P)	out		Right eigenvectors, er(:,k) = r_k.

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compute_face_flux_back_projection

Return the face flux in conservative variables from the reconstructed split fields, F = R (f+ + f-).

Attributes: pure

subroutine compute_face_flux_back_projection(is_characteristic, er, vr, flux)

Arguments

Name	Type	Intent	Attributes	Description
is_characteristic	logical	in		Reconstruction variables: characteristic or conservative.
er	real(kind=R8P)	in		Right eigenvectors, er(:,k) = r_k (unused if conservative).
vr	real(kind=R8P)	in		Reconstructed split fields at the face.
flux	real(kind=R8P)	out		Face flux.

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compute_face_interpolation_fields

Return the fields of the stencil of face i+1/2 to be WENO-interpolated to the face (scheme weno-riemann), in the layout of the Euler body: characteristic w = L q on the Roe average of cells 0 and 1, or primitive (rho, u, v, w, slot), the inert slot in the place of the pressure (P = c0^2 rho needs no separate field).

Attributes: pure

subroutine compute_face_interpolation_fields(gamma, d, S, is_characteristic, qs, qas, fint, er, mu)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2.
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
S	integer(kind=I4P)	in		WENO stencil half-width, S <= S_MAX.
is_characteristic	logical	in		Characteristic (or primitive) variables.
qs	real(kind=R8P)	in		Stencil conservative variables.
qas	real(kind=R8P)	in		Stencil auxiliary variables.
fint	real(kind=R8P)	out		Fields in the WENO upwind layout.
er	real(kind=R8P)	out		Right eigenvectors (unused if primitive).
mu	real(kind=R8P)	out		Magnitude of each field (WENO descaler).

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compute_face_split_fluxes

Project and Lax-Friedrichs-split the stencil of face i+1/2, ready for the WENO upwind reconstruction: the Euler body's algorithm on the isothermal eigensystem (the inert field has speed 0, state 0 and flux 0: it splits to 0).

Attributes: pure

subroutine compute_face_split_fluxes(gamma, d, S, is_characteristic, qs, qas, fsplit, er, mu)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2.
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
S	integer(kind=I4P)	in		WENO stencil half-width, S <= S_MAX.
is_characteristic	logical	in		Characteristic (or conservative) variables.
qs	real(kind=R8P)	in		Stencil conservative variables.
qas	real(kind=R8P)	in		Stencil auxiliary variables.
fsplit	real(kind=R8P)	out		Split fields in the WENO upwind layout.
er	real(kind=R8P)	out		Right eigenvectors (identity if conservative).
mu	real(kind=R8P)	out		Magnitude of each field (WENO descaler).

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compute_face_states

Return the conservative left and right states of face i+1/2 from the WENO-interpolated fields. A state with a non-positive density (or an inadmissible scalar) is replaced by the state of its cell: the face falls back to first order. The pressure follows from the density, so the density is the only thermodynamic condition.

Attributes: pure

subroutine compute_face_states(gamma, is_characteristic, er, vr, q0, q1, qL, qR)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2.
is_characteristic	logical	in		Characteristic (or primitive) variables.
er	real(kind=R8P)	in		Right eigenvectors (unused if primitive).
vr	real(kind=R8P)	in		Interpolated fields.
q0	real(kind=R8P)	in		Conservative variables of cell 0 (left of the face).
q1	real(kind=R8P)	in		Conservative variables of cell 1 (right of the face).
qL	real(kind=R8P)	out		Left state.
qR	real(kind=R8P)	out		Right state.

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compute_flux

Compute the physical flux in direction d: rho u_d (1, u, v, w, 0) + p (0, e_d, 0), p = c0^2 rho.

Attributes: pure

subroutine compute_flux(d, q, qa, f)

Arguments

Name	Type	Intent	Attributes	Description
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
q	real(kind=R8P)	in		Conservative variables.
qa	real(kind=R8P)	in		Auxiliary variables.
f	real(kind=R8P)	out		Physical flux.

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compute_roe_average

Compute the Roe average of two barotropic states: the sqrt(rho)-weighted velocity, the density sqrt(rho_L rho_R), the speed of eos_roe_sound_speed and the pressure of the average density; enthalpy and temperature are set to zero.

Attributes: pure

subroutine compute_roe_average(gamma, qaL, qaR, roe)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		EOS parameter (eos_arg).
qaL	real(kind=R8P)	in		Left state auxiliary variables.
qaR	real(kind=R8P)	in		Right state auxiliary variables.
roe	real(kind=R8P)	out		Roe average.

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compute_riemann_hll

Compute the HLL flux of two isothermal states in direction d with the Einfeldt speeds S_L = min(u_nL - c0, u~_n - c0), S_R = max(u_nR + c0, u~_n + c0) (~ the Roe average); the inert slot's flux is zero.

Attributes: pure

subroutine compute_riemann_hll(gamma, d, qL, qR, f)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2.
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
qL	real(kind=R8P)	in		Left state.
qR	real(kind=R8P)	in		Right state.
f	real(kind=R8P)	out		Flux.

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compute_riemann_hllc

Compute the isothermal HLLC flux of two states in direction d: HLL for the mass and the normal momentum, the transverse momenta and the advected scalars the HLL mass flux times the upwind specific value (the contact, the only wave the isothermal system has between the acoustic ones, carries the tangential velocity and the scalars and nothing else: it needs no star pressure). Checked against eos_oracle.py (flux: consistency, isolated contact).

Attributes: pure

subroutine compute_riemann_hllc(gamma, d, qL, qR, f)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2.
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
qL	real(kind=R8P)	in		Left state.
qR	real(kind=R8P)	in		Right state.
f	real(kind=R8P)	out		Flux.

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compute_riemann_llf

Compute the local Lax-Friedrichs (Rusanov) flux of two states in direction d, F = (f(qL) + f(qR)) / 2 - alpha (qR - qL) / 2, alpha = max(|u_n|) + c0; the inert slot's flux is zero.

Attributes: pure

subroutine compute_riemann_llf(gamma, d, qL, qR, f)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2.
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
qL	real(kind=R8P)	in		Left state.
qR	real(kind=R8P)	in		Right state.
f	real(kind=R8P)	out		Flux.

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conservative_to_auxiliary

Compute the auxiliary variables of a cell from its conservative variables: p = P(rho), a = sqrt(dP/drho), T = P / (rho R), H = 0 (no energy equation).

Attributes: pure

subroutine conservative_to_auxiliary(gamma, R, q, qa)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		EOS parameter (eos_arg).
R	real(kind=R8P)	in		Gas constant.
q	real(kind=R8P)	in		Conservative variables.
qa	real(kind=R8P)	out		Auxiliary variables.

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primitive_to_conservative

Compute the conservative variables of a cell from its primitive variables: the pressure is not a state variable of the isothermal gas (P = c0^2 rho), so p is ignored and the inert slot set to 0 (the advected scalars, if any, are the caller's).

Attributes: pure

subroutine primitive_to_conservative(gamma, r, u, v, w, p, q)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2 (unused).
r	real(kind=R8P)	in		Density.
u	real(kind=R8P)	in		Velocity components.
v	real(kind=R8P)	in		Velocity components.
w	real(kind=R8P)	in		Velocity components.
p	real(kind=R8P)	in		Pressure (ignored).
q	real(kind=R8P)	out		Conservative variables.

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hll_flux

The HLL flux with the Einfeldt speeds, its auxiliary states, physical fluxes and speeds (shared by HLL and HLLC).

Attributes: pure

subroutine hll_flux(gamma, d, qL, qR, qaL, qaR, fL, fR, sL, sR, f)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Isothermal sound speed squared c0^2.
d	integer(kind=I4P)	in		Direction, 1=x, 2=y, 3=z.
qL	real(kind=R8P)	in		Left state.
qR	real(kind=R8P)	in		Right state.
qaL	real(kind=R8P)	out		Left auxiliary variables.
qaR	real(kind=R8P)	out		Right auxiliary variables.
fL	real(kind=R8P)	out		Left physical flux.
fR	real(kind=R8P)	out		Right physical flux.
sL	real(kind=R8P)	out		Left and right wave speeds.
sR	real(kind=R8P)	out		Left and right wave speeds.
f	real(kind=R8P)	out		Flux.

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compute_state_magnitudes

Return the reference magnitude of each conservative variable of a cell, the WENO descaler scale (the Euler body's definition: |rho|, the inert slot |q_5|, every momentum component |rho u|_1 + sqrt(rho p)).

Attributes: pure

subroutine compute_state_magnitudes(q, qa, qm)

Arguments

Name	Type	Intent	Attributes	Description
q	real(kind=R8P)	in		Conservative variables.
qa	real(kind=R8P)	in		Auxiliary variables.
qm	real(kind=R8P)	out		Reference magnitudes.

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Functions

eos_pressure

P(rho) of the table.

Attributes: pure

Returns: real(kind=R8P)

function eos_pressure(gamma, rho) result(p)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Table id.
rho	real(kind=R8P)	in		Density.

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eos_temperature

T = P / (rho R).

Attributes: pure

Returns: real(kind=R8P)

function eos_temperature(gamma, R, rho) result(t)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Table id.
R	real(kind=R8P)	in		Gas constant.
rho	real(kind=R8P)	in		Density.

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eos_sound_speed

a = sqrt(dP/drho).

Attributes: pure

Returns: real(kind=R8P)

function eos_sound_speed(gamma, rho) result(a)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Table id.
rho	real(kind=R8P)	in		Density.

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eos_roe_sound_speed

The speed of the Roe average, the secant sqrt([P] / [rho]): with it the Roe matrix satisfies [F] = A~ [U] for any P(rho). P increases with rho, so the secant is positive, between the two one-sided speeds of a segment and across breaks too. Below a relative density gap of ROE_SECANT_MIN the secant loses digits to cancellation (about eps / gap) while the mid-point speed errs by O(gap^2): there the speed is c(rho mean), both errors below 2e-10.

Attributes: pure

Returns: real(kind=R8P)

function eos_roe_sound_speed(gamma, rhoL, rhoR, pL, pR) result(a)

Arguments

Name	Type	Intent	Attributes	Description
gamma	real(kind=R8P)	in		Table id.
rhoL	real(kind=R8P)	in		Densities.
rhoR	real(kind=R8P)	in		Densities.
pL	real(kind=R8P)	in		Pressures.
pR	real(kind=R8P)	in		Pressures.

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