SalinityTransport Class Reference

Detailed Description

This module contains the solver for the diffusion-advection equation for the transport of salinity.

Author

Nander Wever

#include <SalinityTransport.h>

Public Types
enum	SalinityTransportSolvers { EXPLICIT , IMPLICIT , IMPLICIT2 }

Public Member Functions
	SalinityTransport (size_t nE)
	Class for solving diffusion-advection equation for salinity using the Crank-Nicolson implicit method Solve Richards Equation .
bool	VerifyCFL (const double dt)
	Check for CFL criterion .
bool	VerifyImplicitDt (const double dt)
	Check for Implicit criterion .
bool	SolveSalinityTransportEquationImplicit (const double dt, std::vector< double > &DeltaSal, const double f, const bool DonorCell=true)
	Solve diffusion-advection equation using the Crank-Nicolson implicit, or fully implicit method .
bool	SolveSalinityTransportEquationExplicit (const double dt, std::vector< double > &DeltaSal)
	Solve diffusion-advection equation using the upwind explicit method .

Public Attributes
std::vector< double >	flux_up
std::vector< double >	flux_down
std::vector< double >	flux_up_2
std::vector< double >	flux_down_2
std::vector< double >	dz_
std::vector< double >	dz_up
std::vector< double >	dz_down
std::vector< double >	theta1
std::vector< double >	theta2
std::vector< double >	BrineSal
std::vector< double >	D
std::vector< double >	sb
double	BottomSalinity
double	TopSalinity
double	BottomSalFlux
double	TopSalFlux

Member Enumeration Documentation

SalinityTransportSolvers

enum SalinityTransport::SalinityTransportSolvers

Enumerator
EXPLICIT
IMPLICIT
IMPLICIT2

Constructor & Destructor Documentation

SalinityTransport()

SalinityTransport::SalinityTransport

(

size_t

nE

)

Class for solving diffusion-advection equation for salinity using the Crank-Nicolson implicit method
Solve Richards Equation
.

Author

Nander Wever

Parameters

nE	Domain size (number of elements)

Member Function Documentation

SolveSalinityTransportEquationExplicit()

bool SalinityTransport::SolveSalinityTransportEquationExplicit	(	const double	dt,
		std::vector< double > &	DeltaSal
	)

Solve diffusion-advection equation using the upwind explicit method
.

Author

Nander Wever

Parameters

dt	Time step (s)
DeltaSal	Result vector (change in salinity over time step)

Returns

false on error, true otherwise

SolveSalinityTransportEquationImplicit()

bool SalinityTransport::SolveSalinityTransportEquationImplicit	(	const double	dt,
		std::vector< double > &	DeltaSal,
		const double	f,
		const bool	DonorCell = true
	)

Solve diffusion-advection equation using the Crank-Nicolson implicit, or fully implicit method
.

Author

Nander Wever

This function solves the following equation (n and i denoting time and spatial level, respectively):

\[ \begin{multlined} \frac{ \left ( \theta^{n+1}_i S_{\mathrm{b}, i}^{n+1} - \theta^{n}_i S_{\mathrm{b}, i}^{n} \right ) } { \Delta t } \\ - f \left [ \left ( \frac{ 2 D_{i+1}^{n} \theta^{n+1}_{i+1} S_{\mathrm{b}, i+1}^{n+1} }{ \Delta z_{\mathrm{up}} \left ( \Delta z_{\mathrm{up}} + \Delta z_{\mathrm{down}} \right ) } - \frac{ 2 D_{i}^{n} \theta^{n+1}_{i} S_{\mathrm{b}, i}^{n+1} }{\left ( \Delta z_{\mathrm{up}} \Delta z_{\mathrm{down}} \right ) } + \frac{ D_{i-1}^{n} \theta^{n+1}_{i-1} S_{\mathrm{b}, i-1}^{n+1} }{ \Delta z_{\mathrm{down}} \left ( \Delta z_{\mathrm{up}} + \Delta z_{\mathrm{down}} \right ) } \right ) \right ] \\ - \left ( 1-f \right ) \left [ \left ( \frac{ 2 D_{i+1}^{n} \theta^{n}_{i+1} S_{\mathrm{b}, i+1}^{n} }{ \Delta z_{\mathrm{up}} \left ( \Delta z_{\mathrm{up}} + \Delta z_{\mathrm{down}} \right ) } - \frac{ 2 D_{i}^{n} \theta^{n}_{i} S_{\mathrm{b}, i}^{n} }{\left ( \Delta z_{\mathrm{up}} \Delta z_{\mathrm{down}} \right ) } + \frac{ D_{i-1}^{n} \theta^{n}_{i-1} S_{\mathrm{b}, i-1}^{n} }{ \Delta z_{\mathrm{down}} \left ( \Delta z_{\mathrm{up}} + \Delta z_{\mathrm{down}} \right ) } \right ) \right ] \\ - f \left [ \left ( \frac{q^{n}_{i+1} S_{\mathrm{b},i+1}^{n+1} - q^{n}_{i-1} S_{\mathrm{b},i-1}^{n+1}}{\left ( \Delta z_{\mathrm{up}} + \Delta z_{\mathrm{down}} \right ) } \right ) \right ] - \left ( 1-f \right ) \left [ \left ( \frac{q^{n}_{i+1} S_{\mathrm{b},i+1}^{n} - q^{n}_{i-1} S_{\mathrm{b},i-1}^{n}}{\left ( \Delta z_{\mathrm{up}} + \Delta z_{\mathrm{down}} \right ) } \right ) \right ] - s_{\mathrm{sb}} = 0 \end{multlined} \]

Here, \(f=1\) results in the fully implicit scheme, whereas \(f=0.5\) corresponds to the Crank-Nicolson scheme. The implicit scheme is first order accurate, whereas the Crank-Nicolson scheme is second order accurate. Furthermore, both are unconditionally stable and suffer only minimal numerical diffusion for the advection part. As with many other common schemes, the advection part is not perfectly conserving sharp transitions. Futhermore, the reason to not use the fully implicit or the Crank Nicolson scheme is the occurrence of spurious oscillations in the solution, which negatively impact the accuracy of the simulations more than the negative effect on computational efficiency imposed by the CFL criterion required for the explicit method (see SalinityTransport::SolveSalinityTransportEquationExcplicit).

Parameters

dt	Time step (s)
DeltaSal	Result vector (change in salinity over time step)
f	Set to 0.5 for Crank-Nicolson, or to 1.0 for fully implicit
DonorCell	If true, use mass-conserving donor-cell scheme (upwind). If false, use default implicit discretization

Returns

false on error, true otherwise

VerifyCFL()

bool SalinityTransport::VerifyCFL

(

const double

dt

)

Check for CFL criterion
.

Author

Nander Wever

Parameters

dt	Time step (s)

Returns

true when provided time step dt satisfies CFL criterion, false otherwise.

VerifyImplicitDt()

bool SalinityTransport::VerifyImplicitDt

(

const double

dt

)

Check for Implicit criterion
.

Author

Nander Wever

Parameters

dt	Time step (s)

Returns

true when provided time step dt satisfies criterion to reduce spurious oscillations, false otherwise.

Member Data Documentation

BottomSalFlux

double SalinityTransport::BottomSalFlux

BottomSalinity

double SalinityTransport::BottomSalinity

BrineSal

std::vector<double> SalinityTransport::BrineSal

D

std::vector<double> SalinityTransport::D

dz_

std::vector<double> SalinityTransport::dz_

dz_down

std::vector<double> SalinityTransport::dz_down

dz_up

std::vector<double> SalinityTransport::dz_up

flux_down

std::vector<double> SalinityTransport::flux_down

flux_down_2

std::vector<double> SalinityTransport::flux_down_2

flux_up

std::vector<double> SalinityTransport::flux_up

flux_up_2

std::vector<double> SalinityTransport::flux_up_2

sb

std::vector<double> SalinityTransport::sb

theta1

std::vector<double> SalinityTransport::theta1

theta2

std::vector<double> SalinityTransport::theta2

TopSalFlux

double SalinityTransport::TopSalFlux

TopSalinity

double SalinityTransport::TopSalinity

---

The documentation for this class was generated from the following files:

• SalinityTransport.h
• SalinityTransport.cc