analyticalSolutions

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

Macros
#define	max(a, b)

Functions
int	diffusionSin1D (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Sinusoidal 1D diffusion.
int	diffusionSin2D (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Sinusoidal 2D diffusion.
int	diffusionSin3D (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Sinusoidal 3D diffusion.
int	diffusionSin1D_r (int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *r)
	Sinusoidal 1D diffusion (reaction).
int	diffusionSin2D_r (int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *r)
	Sinusoidal 2D diffusion (reaction).
int	diffusionSin3D_r (int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *r)
	Sinusoidal 3D diffusion (reaction).
int	LinearAD_DiracIC (int *iwork, double *rwork, int nPoints, double T, double *x, double *u)
	Linear Advective Diffusion Dirac Initial Condition.
int	LinearAD_DiracIC_advectiveVelocity (int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
	Linear Advective Diffusion Dirac Initial Condition(advective velocity).
int	LinearAD_DiracIC_diffusiveVelocity (int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
	Linear Advective Diffusion Dirac Initial Condition(diffusive velocity).
int	LinearAD_DiracIC_du (int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
	Linear Advective Diffusion Dirac Initial Condition (du).
int	LinearAD_DiracIC_totalVelocity (int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
	Linear Advective Diffusion Dirac Initial Condition(total velocity).
int	LinearAD_SteadyState (int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
	Linear Advection-Diffusion Steady State.
int	LinearADR_Decay_DiracIC (int *iwork, double *rwork, int nPoints, double T, double *x, double *u)
	Linear Avection Diffusion Reaction Decay Dirac Initial Condition.
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).
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).
int	LinearADR_Sine (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Linear Avection Diffusion Reaction Sine function.
int	LinearADR_Sine_advectiveVelocity (int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
	Linear Avection Diffusion Reaction Sine function (advective velocity).
int	LinearADR_Sine_diffusiveVelocity (int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
	Linear Avection Diffusion Reaction Sine function (diffusive velocity).
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).
int	LinearADR_Sine_du (int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
	Linear Avection Diffusion Reaction Sine function (du).
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).
int	LinearADR_Sine_totalVelocity (int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
	Linear Avection Diffusion Reaction Sine function (total velocity).
int	NonlinearAD_SteadyState (int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
	Nonlinear Advection-Diffusion Steady State.
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.
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).
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).
int	NonlinearDAE (int *iwork, double *rwork, int nPoints, double T, double *x, double *u)
	Nonlinear Differential-algebraic equations.
int	NonlinearDAE_f (int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
	Nonlinear Differential-algebraic equations.
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).
int	PoiseuillePipeFlow (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Poiseuille Flow through a circular pipe.
int	PoiseuillePipeFlow_P (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Poiseuille Flow through a circular pipe.
int	poissonsEquationExp1D (int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
	Poisson Exponential Equation 1D.
int	poissonsEquationExp2D (int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
	Poisson Exponential Equation 2D.
int	poissonsEquationExp3D (int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
	Poisson Exponential Equation 3D.
int	poissonsEquationExp3D_dr (int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *dr)
	Poisson Exponential Equation 3D (dr).
int	poissonsEquationExp1D_r (int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *r)
	Poisson Exponential Equation 1D (reaction).
int	poissonsEquationExp2D_r (int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *r)
	Poisson Exponential Equation 2D (reaction).
int	poissonsEquationExp3D_r (int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *r)
	Poisson Exponential Equation 3D (reaction).
int	STflowSphere_P (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Stokes Flow around moving Sphere.
int	STflowSphere_Vx (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Stokes Flow around moving Sphere.
int	STflowSphere_Vy (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Stokes Flow around moving Sphere.
int	STflowSphere_Vz (int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
	Stokes Flow around moving Sphere.
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).

Detailed Description

Macro Definition Documentation

max

#define max	(		a,
			b )

Value:

(((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 )

extern

Definition at line 2093 of file analyticalSolutions.c.

df()

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

extern

Definition at line 2209 of file analyticalSolutions.c.

diffusionSin1D()

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

extern

Parameters

-	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 )

extern

Parameters

-	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 )

extern

Parameters

-	x-component angular frequency, \( \omega_x \)
iwork	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 )

extern

Parameters

-	x-component angular frequency, \( omega_x \)
iwork	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 )

extern

Parameters

-	x-component angular frequency, \( \omega_x \)
iwork	y-component angular frequency, \( \omega_y \)
rwork	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 )

extern

Parameters

-	x-component angular frequency, \( omega_x \)
iwork	y-component angular frequency, \( omega_y \)
rwork	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 )

extern

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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
u	\( u_0 \)
-	\( x_0[0:2] \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
f	\( u_0 \)
-	\( x_0[0:2] \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
f	\( u_0 \)
-	\( x_0[0:2] \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
f	\( u_0 \)
-	\( x_0[0:2] \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
f	\( u_0 \)
-	\( x_0[0:2] \)
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 )

extern

Parameters

iwork	NOT USED
-	\( b \)
iwork	\( a \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
u	\( u_0 \)
-	\( x_0[0:2] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
u	\( u_0 \)
-	\( x_0[0:2] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
u	\( u_0 \)
-	\( x_0[0:2] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( \omega[0:2] \)
nPoints	\( \omega_0 \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( \omega[0:2] \)
nPoints	\( \omega_0 \)
f	\( b[0:2] \)
-	\( a[0:3,0:3] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( \omega[0:2] \)
nPoints	\( \omega_0 \)
f	\( b[0:2] \)
-	\( a[0:3,0:3] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( \omega[0:2] \)
nPoints	\( \omega_0 \)
u	\( b[0:2] \)
-	\( a[0:3,0:3] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( \omega[0:2] \)
nPoints	\( \omega_0 \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( \omega[0:2] \)
nPoints	\( \omega_0 \)
u	\( b[0:2] \)
-	\( a[0:3,0:3] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( \omega[0:2] \)
nPoints	\( \omega_0 \)
f	\( b[0:2] \)
-	\( a[0:3,0:3] \)
-	\( c \)
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 )

extern

Parameters

iwork	NOT USED
-	\( q \)
iwork	\( r \)
rwork	\( b \)
nPoints	\( a \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
u	\( u_0 \)
-	\( x_0[0:2] \)
-	\( c \)
-	\( d \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
u	\( u_0 \)
-	\( x_0[0:2] \)
-	\( c \)
-	\( d \)
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 )

extern

Parameters

iwork	NOT USED
rwork	\( b[0:2] \)
nPoints	\( n \)
T	\( a \)
x	\( tStart \)
u	\( u_0 \)
-	\( x_0[0:2] \)
-	\( c \)
-	\( d \)
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 )

extern

Parameters

iwork	NOT USED
-	\( a \)
iwork	\( p \)
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 )

extern

Parameters

iwork	NOT USED
-	\( a \)
iwork	\( p \)
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
-	velocity of moving plate in the x-dir, \( vx \)
iwork	width between plates, \( h \)
rwork	x-axis offset, \( xs \)
nPoints	y-axis offset, \( ys \)
t	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 )

extern

Parameters

iwork	NOT USED
-	width between the plates, \( h \)
iwork	viscosity of fluid, \( \mu \)
rwork	the pressure gradient, \( \nabla p \) (neg)
nPoints	the rate of flow per unit width, \( q \)
t	x-axis offset, \( xs \)
x	y-axis offset, \( ys \)
u	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 )

extern

Parameters

iwork	NOT USED
-	radius of pipe, \( R \)
iwork	viscosity of fluid, \( \mu \)
rwork	the pressure gradient, \( \nabla p \) (neg)
nPoints	the rate of flow, \( Q \)
t	x-axis offset, \( xs \)
x	y-axis offset, \( ys \)
u	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 )

extern

Parameters

iwork	NOT USED
-	radius of pipe, \( R \)
iwork	viscosity of fluid, \( \mu \)
rwork	the pressure gradient, \( \nabla p \) (neg)
nPoints	the rate of flow, \( Q \)
nPoints	the length of pipe, \( L \)
t	x-axis offset, \( xs \)
x	y-axis offset, \( ys \)
u	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 )

extern

Parameters

iwork	NOT USED
-	\( K \)
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 )

extern

Parameters

iwork	NOT USED
-	\( K \)
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 )

extern

Parameters

iwork	NOT USED
-	\( K \)
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 )

extern

Parameters

iwork	NOT USED
-	\( K \)
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 )

extern

Parameters

iwork	NOT USED
-	\( K \)
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 )

extern

Parameters

iwork	NOT USED
-	\( K \)
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 )

extern

Parameters

iwork	NOT USED
-	\( K \)
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 )

extern

Parameters

iwork	NOT USED
-	Sphere's x-component of velocity, \( vx \)
iwork	Sphere's y-component of velocity, \( vy \)
rwork	Sphere's z-component of velocity, \( vz \)
nPoints	Sphere's radius, \( rs \)
t	Sphere's center x-component, \( xs \)
x	Sphere's center y-component, \( ys \)
u	Sphere's center z-component, \( zs \)
-	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 )

extern

Parameters

-	Sphere's x-component of velocity, \( vx \)
iwork	Sphere's y-component of velocity, \( vy \)
rwork	Sphere's z-component of velocity, \( vz \)
nPoints	Sphere's radius, \( rs \)
t	Sphere's center x-component, \( xs \)
x	Sphere's center y-component, \( ys \)
u	Sphere's center z-component, \( zs \)
-	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 )

extern

Parameters

iwork	NOT USED
-	Sphere's x-component of velocity, \( vx \)
iwork	Sphere's y-component of velocity, \( vy \)
rwork	Sphere's z-component of velocity, \( vz \)
nPoints	Sphere's radius, \( rs \)
t	Sphere's center x-component, \( xs \)
x	Sphere's center y-component, \( ys \)
u	Sphere's center z-component, \( zs \)
-	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 )

extern

Parameters

iwork	NOT USED
-	Sphere's x-component of velocity, \( vx \)
iwork	Sphere's y-component of velocity, \( vy \)
rwork	Sphere's z-component of velocity, \( vz \)
nPoints	Sphere's radius, \( rs \)
t	Sphere's center x-component, \( xs \)
x	Sphere's center y-component, \( ys \)
u	Sphere's center z-component, \( zs \)
-	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 )

extern

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 )

extern

Definition at line 2181 of file analyticalSolutions.c.

vel()

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

extern

Definition at line 2163 of file analyticalSolutions.c.