A C library of analytical solutions to differential equations for use in verification. More...

Macros
#define	max(a, b) (((a)>(b))?(a):(b))

Functions
int	diffusionSin1D (int iwork, double rwork, int nPoints, double t, double x, double u)
	Sinusoidal 1D diffusion. More...

int	diffusionSin2D (int iwork, double rwork, int nPoints, double t, double x, double u)
	Sinusoidal 2D diffusion. More...

int	diffusionSin3D (int iwork, double rwork, int nPoints, double t, double x, double u)
	Sinusoidal 3D diffusion. More...

int	diffusionSin1D_r (int iwork, double rwork, int nPoints, double t, double x, double u, double *r)
	Sinusoidal 1D diffusion (reaction) More...

int	diffusionSin2D_r (int iwork, double rwork, int nPoints, double t, double x, double u, double *r)
	Sinusoidal 2D diffusion (reaction) More...

int	diffusionSin3D_r (int iwork, double rwork, int nPoints, double t, double x, double u, double *r)
	Sinusoidal 3D diffusion (reaction) More...

int	LinearAD_DiracIC (int iwork, double rwork, int nPoints, double T, double x, double u)
	Linear Advective Diffusion Dirac Initial Condition. More...

int	LinearAD_DiracIC_advectiveVelocity (int iwork, double rwork, int nPoints, double T, double x, double f)
	Linear Advective Diffusion Dirac Initial Condition(advective velocity) More...

int	LinearAD_DiracIC_diffusiveVelocity (int iwork, double rwork, int nPoints, double T, double x, double f)
	Linear Advective Diffusion Dirac Initial Condition(diffusive velocity) More...

int	LinearAD_DiracIC_du (int iwork, double rwork, int nPoints, double T, double x, double f)
	Linear Advective Diffusion Dirac Initial Condition (du) More...

int	LinearAD_DiracIC_totalVelocity (int iwork, double rwork, int nPoints, double T, double x, double f)
	Linear Advective Diffusion Dirac Initial Condition(total velocity) More...

int	LinearAD_SteadyState (int iwork, double rwork, int nPoints, double t, double X, double u)
	Linear Advection-Diffusion Steady State. More...

int	LinearADR_Decay_DiracIC (int iwork, double rwork, int nPoints, double T, double x, double u)
	Linear Avection Diffusion Reaction Decay Dirac Initial Condition. More...

int	LinearADR_Decay_DiracIC_dr (int iwork, double rwork, int nPoints, double T, double x, double u, double *dr)
	Linear Avection Diffusion Reaction Decay Dirac Initial Condition (dr) More...

int	LinearADR_Decay_DiracIC_r (int iwork, double rwork, int nPoints, double T, double x, double u, double *r)
	Linear Avection Diffusion Reaction Decay Dirac Initial Condition (reaction) More...

int	LinearADR_Sine (int iwork, double rwork, int nPoints, double t, double x, double u)
	Linear Avection Diffusion Reaction Sine function. More...

int	LinearADR_Sine_advectiveVelocity (int iwork, double rwork, int nPoints, double t, double x, double f)
	Linear Avection Diffusion Reaction Sine function (advective velocity) More...

int	LinearADR_Sine_diffusiveVelocity (int iwork, double rwork, int nPoints, double t, double x, double f)
	Linear Avection Diffusion Reaction Sine function (diffusive velocity) More...

int	LinearADR_Sine_dr (int iwork, double rwork, int nPoints, double t, double x, double u, double *dr)
	Linear Avection Diffusion Reaction Sine function (dr) More...

int	LinearADR_Sine_du (int iwork, double rwork, int nPoints, double t, double x, double f)
	Linear Avection Diffusion Reaction Sine function (du) More...

int	LinearADR_Sine_r (int iwork, double rwork, int nPoints, double t, double x, double u, double *r)
	Linear Avection Diffusion Reaction Sine function (reaction) More...

int	LinearADR_Sine_totalVelocity (int iwork, double rwork, int nPoints, double t, double x, double f)
	Linear Avection Diffusion Reaction Sine function (total velocity) More...

int	NonlinearAD_SteadyState (int iwork, double rwork, int nPoints, double t, double X, double u)
	Nonlinear Advection-Diffusion Steady State. More...

int	NonlinearADR_Decay_DiracIC (int iwork, double rwork, int nPoints, double T, double x, double u)
	Non Linear Avection Diffusion Reaction Decay Dirac Initial Condition. More...

int	NonlinearADR_Decay_DiracIC_dr (int iwork, double rwork, int nPoints, double T, double x, double u, double *dr)
	Non Linear Avection Diffusion Reaction Decay Dirac Initial Condition (dr) More...

int	NonlinearADR_Decay_DiracIC_r (int iwork, double rwork, int nPoints, double T, double x, double u, double *r)
	Non Linear Avection Diffusion Reaction Decay Dirac Initial Condition (reaction) More...

int	NonlinearDAE (int iwork, double rwork, int nPoints, double T, double x, double u)
	Nonlinear Differential-algebraic equations. More...

int	NonlinearDAE_f (int iwork, double rwork, int nPoints, double t, double x, double f)
	Nonlinear Differential-algebraic equations. More...

int	PlanePoiseuilleFlow_u (int iwork, double rwork, int nPoints, double t, double x, double u)
	Poiseuille Flow between two parallel fixed plates with constant seperation (width). More...

int	PoiseuillePipeFlow (int iwork, double rwork, int nPoints, double t, double x, double u)
	Poiseuille Flow through a circular pipe. More...

int	PoiseuillePipeFlow_P (int iwork, double rwork, int nPoints, double t, double x, double u)
	Poiseuille Flow through a circular pipe. More...

int	poissonsEquationExp1D (int iwork, double rwork, int nPoints, double t, double X, double u)
	Poisson Exponential Equation 1D. More...

int	poissonsEquationExp2D (int iwork, double rwork, int nPoints, double t, double X, double u)
	Poisson Exponential Equation 2D. More...

int	poissonsEquationExp3D (int iwork, double rwork, int nPoints, double t, double X, double u)
	Poisson Exponential Equation 3D. More...

int	poissonsEquationExp3D_dr (int iwork, double rwork, int nPoints, double t, double X, double u, double *dr)
	Poisson Exponential Equation 3D (dr) More...

int	poissonsEquationExp1D_r (int iwork, double rwork, int nPoints, double t, double X, double u, double *r)
	Poisson Exponential Equation 1D (reaction) More...

int	poissonsEquationExp2D_r (int iwork, double rwork, int nPoints, double t, double X, double u, double *r)
	Poisson Exponential Equation 2D (reaction) More...

int	poissonsEquationExp3D_r (int iwork, double rwork, int nPoints, double t, double X, double u, double *r)
	Poisson Exponential Equation 3D (reaction) More...

int	STflowSphere_P (int iwork, double rwork, int nPoints, double t, double x, double u)
	Stokes Flow around moving Sphere. More...

int	STflowSphere_Vx (int iwork, double rwork, int nPoints, double t, double x, double u)
	Stokes Flow around moving Sphere. More...

int	STflowSphere_Vy (int iwork, double rwork, int nPoints, double t, double x, double u)
	Stokes Flow around moving Sphere. More...

int	STflowSphere_Vz (int iwork, double rwork, int nPoints, double t, double x, double u)
	Stokes Flow around moving Sphere. More...

void	coords (double vx, double vy, double vz, double xS, double yS, double zS, double x, double r, double theta, double norm_v, double eR, double eTHETA)

void	vel (double rS, double norm_v, double r, double theta, double vR, double vTHETA)

double	uOfX_df (double nlC, double lu)

double	uOfX_f (double a, double b, double nlC, double nlD, double x, double lu)

double	f (double C, double b, double a, int q, int r)

double	df (double C, double b, double a, int q, int r)

int	PlaneCouetteFlow_u (int iwork, double rwork, int nPoints, double t, double x, double u)
	Couette Flow between two parallel plates. One moving relative to the other with constant seperation (width). More...

Detailed Description

Macro Definition Documentation

◆ max

#define max	(	a,
		b
	)	(((a)>(b))?(a):(b))

Definition at line 14 of file analyticalSolutions.h.

Function Documentation

◆ coords()

void coords	(	double	vx,
		double	vy,
		double	vz,
		double	xS,
		double	yS,
		double	zS,
		double *	x,
		double *	r,
		double *	theta,
		double *	norm_v,
		double *	eR,
		double *	eTHETA
	)

Definition at line 2093 of file analyticalSolutions.c.

◆ df()

double df	(	double	C,
		double	b,
		double	a,
		int	q,
		int	r
	)

Definition at line 2209 of file analyticalSolutions.c.

◆ diffusionSin1D()

int diffusionSin1D	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork[0]	angular frequency, \( \omega_x \)
rwork	NOT USED
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[ -\Delta u = f \]

\[ f = -(2 \pi \omega_x)^2 \sin(2 \pi \omega_x x) \]

Returns: status code

Definition at line 84 of file analyticalSolutions.c.

◆ diffusionSin1D_r()

int diffusionSin1D_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u,
		double *	r
	)

Parameters

iwork[0]	x-component angular frequency, \( \omega_x \)
rwork	NOT USED
nPoints	- total number of points
t	NOT USED
x	input array
u	NOT USED
r	output array

\[ -\Delta u = -r \]

\[ r = (2 \pi \omega_x)^2 \sin(2 \pi \omega_x x) \]

Returns: status code

Definition at line 172 of file analyticalSolutions.c.

◆ diffusionSin2D()

int diffusionSin2D	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork[0]	x-component angular frequency, \( \omega_x \)
iwork[1]	y-component angular frequency, \( \omega_y \)
rwork	NOT USED
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[ -\Delta u = f \]

\[ f = -(2\pi\omega_x)^2\sin(2 \pi \omega_x x) -(2 \pi \omega_y)^2\sin(2 \pi \omega_y y)\]

Returns: status code

Definition at line 111 of file analyticalSolutions.c.

◆ diffusionSin2D_r()

int diffusionSin2D_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u,
		double *	r
	)

Parameters

iwork[0]	x-component angular frequency, \( omega_x \)
iwork[1]	y-component angular frequency, \( omega_y \)
rwork	NOT USED
nPoints	- total number of points
t	NOT USED
x	input array
u	NOT USED
r	output array

\[-\Delta u = -r \]

\[r= (2\pi\omega_x)^2\sin(2 \pi \omega_x x)+(2 \pi \omega_y)^2\sin(2 \pi \omega_y y)\]

Returns: status code

Definition at line 200 of file analyticalSolutions.c.

◆ diffusionSin3D()

int diffusionSin3D	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork[0]	x-component angular frequency, \( \omega_x \)
iwork[1]	y-component angular frequency, \( \omega_y \)
iwork[2]	z-component angular frequency, \( \omega_z \)
rwork	NOT USED
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[ -\Delta u = f \]

\[ f = -(2\pi\omega_x)^2\sin(2\pi\omega_x x) -(2\pi\omega_y)^2\sin(2\pi\omega_y y)-(2\pi\omega_z)^2sin(2 \pi \omega_z z)\]

Returns: status code

Definition at line 141 of file analyticalSolutions.c.

◆ diffusionSin3D_r()

int diffusionSin3D_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u,
		double *	r
	)

Parameters

iwork[0]	x-component angular frequency, \( omega_x \)
iwork[1]	y-component angular frequency, \( omega_y \)
iwork[2]	z-component angular frequency, \( omega_z \)
rwork	NOT USED
nPoints	- total number of points
t	NOT USED
x	input array
u	NOT USED
r	output array

\[-\Delta u = -r \]

\[r=(2\pi\omega_x)^2\sin(2\pi\omega_x x) +(2\pi\omega_y)^2\sin(2\pi\omega_y y)+(2\pi\omega_z)^2sin(2 \pi \omega_z z)\]

Returns: status code

Definition at line 231 of file analyticalSolutions.c.

◆ f()

double f	(	double	C,
		double	b,
		double	a,
		int	q,
		int	r
	)

Definition at line 2185 of file analyticalSolutions.c.

◆ LinearAD_DiracIC()

int LinearAD_DiracIC	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

The exact solution of

\[ u_t + \nabla \cdot (b u - a \nabla u) = 0\]

on an infinite domain with dirac initial data

\[u_0 = \int u_0 \delta(x - x_0)\]

also returns advective, diffusive, and total flux

Returns: status code

Definition at line 273 of file analyticalSolutions.c.

◆ LinearAD_DiracIC_advectiveVelocity()

int LinearAD_DiracIC_advectiveVelocity	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array, vector

The exact solution of

\[ u_t + \nabla \cdot (b u - a \nabla u) = 0\]

on an infinite domain with dirac initial data

\[u_0 = \int u_0 \delta(x - x_0)\]

Returns: status code

Definition at line 337 of file analyticalSolutions.c.

◆ LinearAD_DiracIC_diffusiveVelocity()

int LinearAD_DiracIC_diffusiveVelocity	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array, vector

The exact solution of

\[ u_t + \nabla \cdot (b u - a \nabla u) = 0\]

on an infinite domain with dirac initial data

\[u_0 = \int u_0 \delta(x - x_0)\]

Returns: status code

Definition at line 381 of file analyticalSolutions.c.

◆ LinearAD_DiracIC_du()

int LinearAD_DiracIC_du	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array, vector

The exact solution of

\[ u_t + \nabla \cdot (b u - a \nabla u) = 0\]

on an infinite domain with dirac initial data

\[u_0 = \int u_0 \delta(x - x_0)\]

also returns advective, diffusive, and total flux

Returns: status code

Definition at line 423 of file analyticalSolutions.c.

◆ LinearAD_DiracIC_totalVelocity()

int LinearAD_DiracIC_totalVelocity	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array, vector

The exact solution of

\[ u_t + \nabla \cdot (b u - a \nabla u) = 0\]

on an infinite domain with dirac initial data

\[u_0 = \int u_0 \delta(x - x_0)\]

Returns: status code

Definition at line 471 of file analyticalSolutions.c.

◆ LinearAD_SteadyState()

int LinearAD_SteadyState	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	\( b \)
rwork[1]	\( a \)
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[(bu - au_x)_x = 0\]

\[ u(0) = 1\]

\[ u(1) = 0\]

Returns: status code

Definition at line 508 of file analyticalSolutions.c.

◆ LinearADR_Decay_DiracIC()

int LinearADR_Decay_DiracIC	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
rwork[10]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

The exact solution of

\f[ u_t + \nabla \cdot(bu - a \nabla u) + cu= 0 \f]

on an infinite domain with Dirac initial data.
Also returns the fluxes (by inheritance).

Returns: status code

Definition at line 566 of file analyticalSolutions.c.

◆ LinearADR_Decay_DiracIC_dr()

int LinearADR_Decay_DiracIC_dr	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u,
		double *	dr
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
rwork[10]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array
dr	output array

The exact solution of

\f[ u_t + \nabla \cdot(bu - a \nabla u) + cu= 0 \f]

on an infinite domain with Dirac initial data.
Also returns the fluxes (by inheritance).

Returns: status code

Definition at line 613 of file analyticalSolutions.c.

◆ LinearADR_Decay_DiracIC_r()

int LinearADR_Decay_DiracIC_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u,
		double *	r
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
rwork[10]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array
r	output array

The exact solution of

\f[ u_t + \nabla \cdot(bu - a \nabla u) + cu= 0 \f]

on an infinite domain with Dirac initial data.
Also returns the fluxes (by inheritance).

Returns: status code

Definition at line 652 of file analyticalSolutions.c.

◆ LinearADR_Sine()

int LinearADR_Sine	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( \omega[0:2] \)
rwork[3]	\( \omega_0 \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

An exact solution and source term for

\[ \nabla \cdot (\vec b u - \ddot a \nabla u) + c u + d = 0 \]

where

\[ u(x) = sin(\vec \omega \cdot \vec x + \omega_0) = sin(Ax - b)\]

\[r(u,x) = - ((\ddot a \vec \omega) \cdot \vec \omega) u - (\vec b \cdot \vec \omega) cos(Ax - b)\]

\[ = c u + D cos(Ax - b) \]

\[ = cu + d \]

also returns the advective, diffusive, and total velocity at a point.

Returns: status code

Definition at line 691 of file analyticalSolutions.c.

◆ LinearADR_Sine_advectiveVelocity()

int LinearADR_Sine_advectiveVelocity	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( \omega[0:2] \)
rwork[3]	\( \omega_0 \)
rwork[4:6]	\( b[0:2] \)
rwork[7:15]	\( a[0:3,0:3] \)
rwork[16]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array (vector)

An exact solution and source term for

\[ \nabla \cdot (\vec b u - \ddot a \nabla u) + c u + d = 0 \]

where

\[ u(x) = sin(\vec \omega \cdot \vec x + \omega_0) = sin(Ax - b)\]

\[r(u,x) = - ((\ddot a \vec \omega) \cdot \vec \omega) u - (\vec b \cdot \vec \omega) cos(Ax - b)\]

\[ = c u + D cos(Ax - b) \]

\[ = cu + d \]

returns the advective velocity at a point.

Returns: status code

Definition at line 749 of file analyticalSolutions.c.

◆ LinearADR_Sine_diffusiveVelocity()

int LinearADR_Sine_diffusiveVelocity	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( \omega[0:2] \)
rwork[3]	\( \omega_0 \)
rwork[4:6]	\( b[0:2] \)
rwork[7:15]	\( a[0:3,0:3] \)
rwork[16]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array (vector)

An exact solution and source term for

\[ \nabla \cdot (\vec b u - \ddot a \nabla u) + c u + d = 0 \]

where

\[ u(x) = sin(\vec \omega \cdot \vec x + \omega_0) = sin(Ax - b)\]

\[r(u,x) = - ((\ddot a \vec \omega) \cdot \vec \omega) u - (\vec b \cdot \vec \omega) cos(Ax - b)\]

\[ = c u + D cos(Ax - b) \]

\[ = cu + d \]

returns the diffusive velocity at a point.

Returns: status code

Definition at line 802 of file analyticalSolutions.c.

◆ LinearADR_Sine_dr()

int LinearADR_Sine_dr	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u,
		double *	dr
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( \omega[0:2] \)
rwork[3]	\( \omega_0 \)
rwork[4:6]	\( b[0:2] \)
rwork[7:15]	\( a[0:3,0:3] \)
rwork[16]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array
dr	output array

An exact solution and source term for

\[ \nabla \cdot (\vec b u - \ddot a \nabla u) + c u + d = 0 \]

where

\[ u(x) = sin[\vec \omega \cdot \vec x + \omega_0) ] = sin(Ax - b) \f \f[r(u,x) = - ((\ddot a \vec \omega) \cdot \vec \omega) u - (\vec b \cdot \vec \omega) cos(Ax - b)\]

\[ = c u + D cos(Ax - b) \]

\[ = cu + d \]

also returns the advective, diffusive, and total velocity at a point.

Returns: status code

Definition at line 851 of file analyticalSolutions.c.

◆ LinearADR_Sine_du()

int LinearADR_Sine_du	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( \omega[0:2] \)
rwork[3]	\( \omega_0 \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array (vector)

An exact solution and source term for

\[ \nabla \cdot (\vec b u - \ddot a \nabla u) + c u + d = 0 \]

where

\[ u(x) = sin(\vec \omega \cdot \vec x + \omega_0) = sin(Ax - b)\]

\[r(u,x) = - ((\ddot a \vec \omega) \cdot \vec \omega) u - (\vec b \cdot \vec \omega) cos(Ax - b)\]

\[ = c u + D cos(Ax - b) \]

\[ = cu + d \]

also returns the advective, diffusive, and total velocity at a point.

Returns: status code

Definition at line 890 of file analyticalSolutions.c.

◆ LinearADR_Sine_r()

int LinearADR_Sine_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u,
		double *	r
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( \omega[0:2] \)
rwork[3]	\( \omega_0 \)
rwork[4:6]	\( b[0:2] \)
rwork[7:15]	\( a[0:3,0:3] \)
rwork[16]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array
r	output array

An exact solution and source term for

\[ \nabla \cdot (\vec b u - \ddot a \nabla u) + c u + d = 0 \]

where

\[ u(x) = sin(\vec \omega \cdot \vec x + \omega_0) = sin(Ax - b)\]

\[r(u,x) = - ((\ddot a \vec \omega) \cdot \vec \omega) u - (\vec b \cdot \vec \omega) cos(Ax - b)\]

\[ = c u + D cos(Ax - b) \]

\[ = cu + d \]

also returns the advective, diffusive, and total velocity at a point.

Returns: status code

Definition at line 938 of file analyticalSolutions.c.

◆ LinearADR_Sine_totalVelocity()

int LinearADR_Sine_totalVelocity	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( \omega[0:2] \)
rwork[3]	\( \omega_0 \)
rwork[4:6]	\( b[0:2] \)
rwork[7:15]	\( a[0:3,0:3] \)
rwork[16]	\( c \)
nPoints	total number of points
t	NOT USED
x	input array
f	output array (vector)

An exact solution and source term for

\[ \nabla \cdot (\vec b u - \ddot a \nabla u) + c u + d = 0 \]

where

\[ u(x) = sin(\vec \omega \cdot \vec x + \omega_0) = sin(Ax - b)\]

\[r(u,x) = - ((\ddot a \vec \omega) \cdot \vec \omega) u - (\vec b \cdot \vec \omega) cos(Ax - b)\]

\[ = c u + D cos(Ax - b) \]

\[ = cu + d \]

returns the total velocity at a point.

Returns: status code

Definition at line 997 of file analyticalSolutions.c.

◆ NonlinearAD_SteadyState()

int NonlinearAD_SteadyState	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	\( q \)
rwork[1]	\( r \)
rwork[2]	\( b \)
rwork[3]	\( a \)
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[(bu^q - a(u^r)_x)_x = 0\]

\[ u(0) = 1\]

\[ u(1) = 0\]

Returns: status code

Definition at line 1036 of file analyticalSolutions.c.

◆ NonlinearADR_Decay_DiracIC()

int NonlinearADR_Decay_DiracIC	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
rwork[10]	\( c \)
rwork[11]	\( d \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array
dr	output array

The approximate analytical solution of

\f[ u_t + \nabla \cdot(bu - a \nabla u) + cu^d= 0 \f]

on an infinite domain with Dirac initial data.
Also returns the fluxes (by inheritance).

Returns: status code

Definition at line 1164 of file analyticalSolutions.c.

◆ NonlinearADR_Decay_DiracIC_dr()

int NonlinearADR_Decay_DiracIC_dr	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u,
		double *	dr
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
rwork[10]	\( c \)
rwork[11]	\( d \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array
dr	output array

The approximate analytical solution of

\f[ u_t + \nabla \cdot(bu - a \nabla u) + cu^d= 0 \f]

on an infinite domain with Dirac initial data.
Also returns the fluxes (by inheritance).

Returns: status code

Definition at line 1216 of file analyticalSolutions.c.

◆ NonlinearADR_Decay_DiracIC_r()

int NonlinearADR_Decay_DiracIC_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u,
		double *	r
	)

Parameters

iwork	NOT USED
rwork[0:2]	\( b[0:2] \)
rwork[3]	\( n \)
rwork[4]	\( a \)
rwork[5]	\( tStart \)
rwork[6]	\( u_0 \)
rwork[7:9]	\( x_0[0:2] \)
rwork[10]	\( c \)
rwork[11]	\( d \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array
r	output array

The approximate analytical solution of

\f[ u_t + \nabla \cdot(bu - a \nabla u) + cu^d= 0 \f]

on an infinite domain with Dirac initial data.
Also returns the fluxes (by inheritance).

Returns: status code

Definition at line 1257 of file analyticalSolutions.c.

◆ NonlinearDAE()

int NonlinearDAE	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	T,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	\( a \)
rwork[1]	\( p \)
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[u_t = - a*max(u,0)^p\]

\[u(0) = 1\]

Returns: status code

Definition at line 1287 of file analyticalSolutions.c.

◆ NonlinearDAE_f()

int NonlinearDAE_f	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	f
	)

Parameters

iwork	NOT USED
rwork[0]	\( a \)
rwork[1]	\( p \)
nPoints	- total number of points
t	NOT USED
x	input array
f	output array, vector

\[u_t = - a*max(u,0)^p\]

\[u(0) = 1\]

Returns: status code

Definition at line 1330 of file analyticalSolutions.c.

◆ PlaneCouetteFlow_u()

int PlaneCouetteFlow_u	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	velocity of moving plate in the x-dir, \( vx \)
rwork[1]	width between plates, \( h \)
rwork[2]	x-axis offset, \( xs \)
rwork[3]	y-axis offset, \( ys \)
rwork[4]	z-axis offset, \( zs \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

x-momentum equation

\[ \nabla^2 u_x = 0 \]

\[ u(y) = \frac{vx}{h} * y\]

\[ u(0) = 0 \]

\[ u(h) = vx \]

Axis of origin: at bottom plate

Returns: status code

Definition at line 36 of file analyticalSolutions.c.

◆ PlanePoiseuilleFlow_u()

int PlanePoiseuilleFlow_u	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	width between the plates, \( h \)
rwork[1]	viscosity of fluid, \( \mu \)
rwork[2]	the pressure gradient, \( \nabla p \) (neg)
rwork[3]	the rate of flow per unit width, \( q \)
rwork[4]	x-axis offset, \( xs \)
rwork[5]	y-axis offset, \( ys \)
rwork[6]	z-axis offset, \( zs \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

x-momentum equation

\[ (\nabla p + \mu \nabla^2 u)_x = 0 \]

\[ u(y) = 4 u_{max} y (\frac{h-y}{h^2}) \]

\[ u(0) = u(h) = 0 \]

\[ u(h/2) = u_{max} \]

\[u_{max} = \frac{-\nabla p}{8 \mu} h^2 = \frac{3}{2} \frac{q}{h} \]

\( u_{max} \) = maximum velocity at the centerline
Axis of origin: at bottom plate

Returns: status code

Definition at line 1380 of file analyticalSolutions.c.

◆ PoiseuillePipeFlow()

int PoiseuillePipeFlow	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	radius of pipe, \( R \)
rwork[1]	viscosity of fluid, \( \mu \)
rwork[2]	the pressure gradient, \( \nabla p \) (neg)
rwork[3]	the rate of flow, \( Q \)
rwork[4]	x-axis offset, \( xs \)
rwork[5]	y-axis offset, \( ys \)
rwork[6]	z-axis offset, \( zs \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

x-momentum equation

\[ (\nabla p + \mu \nabla^2 u)_x = 0 \]

\[ u(r) = u_{max} (1 - \frac{r^2}{R^2}) \]

\[ u(0) = u_{max} \]

\[ u(R) = 0 \]

\[ u_{max} = -\nabla p \frac{R^2}{4 \mu}= \frac{2Q}{\pi R^2}\]

\( u_{max} \) = maximum velocity at the centerline
Axis of origin: at center of pipe

Returns: status code

Definition at line 1466 of file analyticalSolutions.c.

◆ PoiseuillePipeFlow_P()

int PoiseuillePipeFlow_P	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	radius of pipe, \( R \)
rwork[1]	viscosity of fluid, \( \mu \)
rwork[2]	the pressure gradient, \( \nabla p \) (neg)
rwork[3]	the rate of flow, \( Q \)
rwork[3]	the length of pipe, \( L \)
rwork[4]	x-axis offset, \( xs \)
rwork[5]	y-axis offset, \( ys \)
rwork[6]	z-axis offset, \( zs \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

\[ p - p_0 = -\nabla p L(1-x)\]

\( p_0 =p(L)\) = 0, the atmospheric pressure at the outflow
Axis of origin: at center of pipe

Returns: status code

Definition at line 1554 of file analyticalSolutions.c.

◆ poissonsEquationExp1D()

int poissonsEquationExp1D	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	\( K \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

\[-u_{xx} - f = 0\]

\[u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}\]

\[f = -K \{[y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+ [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+ [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]\}e^{x^2 + y^2 + z^2}\]

Returns: status code

Definition at line 1620 of file analyticalSolutions.c.

◆ poissonsEquationExp1D_r()

int poissonsEquationExp1D_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u,
		double *	r
	)

Parameters

iwork	NOT USED
rwork[0]	\( K \)
nPoints	total number of points
t	NOT USED
x	input array
u	NOT USED
r	output array

\[-u_{xx} - f = 0\]

\[u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}\]

\[f = -K \{[y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+ [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+ [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]\}e^{x^2 + y^2 + z^2}\]

Returns: status code

Definition at line 1754 of file analyticalSolutions.c.

◆ poissonsEquationExp2D()

int poissonsEquationExp2D	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	\( K \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

\[-u_{xx} - f = 0\]

\[u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}\]

\[f = -K \{[y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+ [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+ [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]\}e^{x^2 + y^2 + z^2}\]

Returns: status code

Definition at line 1659 of file analyticalSolutions.c.

◆ poissonsEquationExp2D_r()

int poissonsEquationExp2D_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u,
		double *	r
	)

Parameters

iwork	NOT USED
rwork[0]	\( K \)
nPoints	total number of points
t	NOT USED
x	input array
u	NOT USED
r	output array

\[-u_{xx} - f = 0\]

\[u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}\]

\[f = -K \{[y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+ [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+ [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]\}e^{x^2 + y^2 + z^2}\]

Returns: status code

Definition at line 1787 of file analyticalSolutions.c.

◆ poissonsEquationExp3D()

int poissonsEquationExp3D	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	\( K \)
nPoints	total number of points
t	NOT USED
x	input array
u	output array

\[-u_{xx} - f = 0\]

\[u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}\]

\[f = -K \{[y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+ [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+ [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]\}e^{x^2 + y^2 + z^2}\]

Returns: status code

Definition at line 1690 of file analyticalSolutions.c.

◆ poissonsEquationExp3D_dr()

int poissonsEquationExp3D_dr	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u,
		double *	dr
	)

Parameters

iwork	NOT USED
rwork[0]	\( K \)
nPoints	total number of points
t	NOT USED
x	input array
u	NOT USED
dr	output array

\[-u_{xx} - f = 0\]

\[u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}\]

\[f = -K \{[y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+ [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+ [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]\}e^{x^2 + y^2 + z^2}\]

Returns: status code

Definition at line 1724 of file analyticalSolutions.c.

◆ poissonsEquationExp3D_r()

int poissonsEquationExp3D_r	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	X,
		double *	u,
		double *	r
	)

Parameters

iwork	NOT USED
rwork[0]	\( K \)
nPoints	total number of points
t	NOT USED
x	input array
u	NOT USED
r	output array

\[-u_{xx} - f = 0\]

\[u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}\]

\[f = -K \{[y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+ [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+ [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]\}e^{x^2 + y^2 + z^2}\]

Returns: status code

Definition at line 1822 of file analyticalSolutions.c.

◆ STflowSphere_P()

int STflowSphere_P	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	Sphere's x-component of velocity, \( vx \)
rwork[1]	Sphere's y-component of velocity, \( vy \)
rwork[2]	Sphere's z-component of velocity, \( vz \)
rwork[3]	Sphere's radius, \( rs \)
rwork[4]	Sphere's center x-component, \( xs \)
rwork[5]	Sphere's center y-component, \( ys \)
rwork[6]	Sphere's center z-component, \( zs \)
rwork[7]	viscosity of fluid, \( mu \)
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[ \nabla p = \mu \nabla^2 u \]

\[ p = \frac{(-3 * rs * \mu \| \vec v\| cos\theta)}{2r^2}\]

Returns: status code

Definition at line 1864 of file analyticalSolutions.c.

◆ STflowSphere_Vx()

int STflowSphere_Vx	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

rwork[0]	Sphere's x-component of velocity, \( vx \)
rwork[1]	Sphere's y-component of velocity, \( vy \)
rwork[2]	Sphere's z-component of velocity, \( vz \)
rwork[3]	Sphere's radius, \( rs \)
rwork[4]	Sphere's center x-component, \( xs \)
rwork[5]	Sphere's center y-component, \( ys \)
rwork[6]	Sphere's center z-component, \( zs \)
rwork[7]	viscosity of fluid, \( mu \)
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[ \nabla p = \mu \nabla^2 u \]

\[ v^r = \|\vec v \| cos\theta(1-\frac{3rs}{2r} + \frac{rs^3}{2r^3})\]

\[ v^\theta = -\|\vec v \| sin\theta(1-\frac{3rs}{4r} - \frac{rs^3}{4r^3})\]

\[v^r(r>rs) = 0\]

\[v^\theta(r>rs) = 0\]

\[e^R = \frac{\vec x}{\|\vec x \|} ; \{ x_x = x - xs;x_y = y-ys; x_z = z-zs\} \]

\[e^\theta = (e^R - \frac{\vec v}{\|\vec v \|}) - [(e^R - \frac{\vec v}{\|\vec v \|}) \cdot e^R]e^R\]

\[u_x = v^r e^R_x + v^\theta e^\theta_x \]

Returns: status code

Definition at line 1929 of file analyticalSolutions.c.

◆ STflowSphere_Vy()

int STflowSphere_Vy	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	Sphere's x-component of velocity, \( vx \)
rwork[1]	Sphere's y-component of velocity, \( vy \)
rwork[2]	Sphere's z-component of velocity, \( vz \)
rwork[3]	Sphere's radius, \( rs \)
rwork[4]	Sphere's center x-component, \( xs \)
rwork[5]	Sphere's center y-component, \( ys \)
rwork[6]	Sphere's center z-component, \( zs \)
rwork[7]	viscosity of fluid, \( mu \)
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[ \nabla p = \mu \nabla^2 u \]

\[ v^r = \|\vec v \| cos\theta(1-\frac{3rs}{2r} + \frac{rs^3}{2r^3})\]

\[ v^\theta = -\|\vec v \| sin\theta(1-\frac{3rs}{4r} - \frac{rs^3}{4r^3})\]

\[v^r(r>rs) = 0\]

\[v^\theta(r>rs) = 0\]

\[e^R = \frac{\vec x}{\|\vec x \|} ; \{ x_x = x - xs;x_y = y-ys; x_z = z-zs\} \]

\[e^\theta = (e^R - \frac{\vec v}{\|\vec v \|}) - [(e^R - \frac{\vec v}{\|\vec v \|}) \cdot e^R]e^R\]

\[u_y = v^r e^R_y + v^\theta e^\theta_y \]

Returns: status code

Definition at line 1992 of file analyticalSolutions.c.

◆ STflowSphere_Vz()

int STflowSphere_Vz	(	int *	iwork,
		double *	rwork,
		int	nPoints,
		double	t,
		double *	x,
		double *	u
	)

Parameters

iwork	NOT USED
rwork[0]	Sphere's x-component of velocity, \( vx \)
rwork[1]	Sphere's y-component of velocity, \( vy \)
rwork[2]	Sphere's z-component of velocity, \( vz \)
rwork[3]	Sphere's radius, \( rs \)
rwork[4]	Sphere's center x-component, \( xs \)
rwork[5]	Sphere's center y-component, \( ys \)
rwork[6]	Sphere's center z-component, \( zs \)
rwork[7]	viscosity of fluid, \( mu \)
nPoints	- total number of points
t	NOT USED
x	input array
u	output array

\[ \nabla p = \mu \nabla^2 u \]

\[ v^r = \|\vec v \| cos\theta(1-\frac{3rs}{2r} + \frac{rs^3}{2r^3})\]

\[ v^\theta = -\|\vec v \| sin\theta(1-\frac{3rs}{4r} - \frac{rs^3}{4r^3})\]

\[v^r(r>rs) = 0\]

\[v^\theta(r>rs) = 0\]

\[e^R = \frac{\vec x}{\|\vec x \|} ; \{ x_x = x - xs;x_y = y-ys; x_z = z-zs\} \]

\[e^\theta = (e^R - \frac{\vec v}{\|\vec v \|}) - [(e^R - \frac{\vec v}{\|\vec v \|}) \cdot e^R]e^R\]

\[u_z = v^r e^R_z + v^\theta e^\theta_z \]

Returns: status code

Definition at line 2055 of file analyticalSolutions.c.

◆ uOfX_df()

double uOfX_df	(	double	nlC,
		double	lu
	)

Definition at line 2177 of file analyticalSolutions.c.

◆ uOfX_f()

double uOfX_f	(	double	a,
		double	b,
		double	nlC,
		double	nlD,
		double	x,
		double	lu
	)

Definition at line 2181 of file analyticalSolutions.c.

◆ vel()

void vel	(	double	rS,
		double	norm_v,
		double	r,
		double	theta,
		double *	vR,
		double *	vTHETA
	)

Definition at line 2163 of file analyticalSolutions.c.

Macros

Functions

Detailed Description

Macro Definition Documentation

◆ max

Function Documentation

◆ coords()

◆ df()

◆ diffusionSin1D()

◆ diffusionSin1D_r()

◆ diffusionSin2D()

◆ diffusionSin2D_r()

◆ diffusionSin3D()

◆ diffusionSin3D_r()

◆ f()

◆ LinearAD_DiracIC()

◆ LinearAD_DiracIC_advectiveVelocity()

◆ LinearAD_DiracIC_diffusiveVelocity()

◆ LinearAD_DiracIC_du()

◆ LinearAD_DiracIC_totalVelocity()

◆ LinearAD_SteadyState()

◆ LinearADR_Decay_DiracIC()

◆ LinearADR_Decay_DiracIC_dr()

◆ LinearADR_Decay_DiracIC_r()

◆ LinearADR_Sine()

◆ LinearADR_Sine_advectiveVelocity()

◆ LinearADR_Sine_diffusiveVelocity()

◆ LinearADR_Sine_dr()

◆ LinearADR_Sine_du()

◆ LinearADR_Sine_r()

◆ LinearADR_Sine_totalVelocity()

◆ NonlinearAD_SteadyState()

◆ NonlinearADR_Decay_DiracIC()

◆ NonlinearADR_Decay_DiracIC_dr()

◆ NonlinearADR_Decay_DiracIC_r()

◆ NonlinearDAE()

◆ NonlinearDAE_f()

◆ PlaneCouetteFlow_u()

◆ PlanePoiseuilleFlow_u()

◆ PoiseuillePipeFlow()

◆ PoiseuillePipeFlow_P()

◆ poissonsEquationExp1D()

◆ poissonsEquationExp1D_r()

◆ poissonsEquationExp2D()

◆ poissonsEquationExp2D_r()

◆ poissonsEquationExp3D()

◆ poissonsEquationExp3D_dr()

◆ poissonsEquationExp3D_r()

◆ STflowSphere_P()

◆ STflowSphere_Vx()

◆ STflowSphere_Vy()

◆ STflowSphere_Vz()

◆ uOfX_df()

◆ uOfX_f()

◆ vel()