Physics of Lakes pp 91-127 | Cite as

# Comparing Numerical Methods for Convectively-Dominated Problems

## Abstract

In this chapter the primitive fundamentals of numerical techniques are assumed known. Its content is classical insofar that the significant scientific numerical research work has been published in the second half of the twentieth century, largely before the 1990s. The content is certainly known to numerical analysts. However, to the physical limnologist not specialized in the computational determination of convective-diffusive processes, the results for the flow of advected material requires subtle discretization to correctly predict the physical flow processes. The recognition of this fact is the reason for the presentation of the various outlined discretization features that are known to the numerical specialist. It is shown that in three-dimensional modeling of baroclinic wind-induced flows the convection terms take on considerable importance; if not in the equations of motion, then certainly in the temperature and salinity equations. Besides, the interest in hydrodynamic modeling as a tool to study water quality problems led to the use of convection-diffusion equations and their approximate treatment to simulate transports of dissolved or suspended matter in natural basins. To our surprise, so far, in computational lake and ocean dynamics, only a few models use high-resolution schemes to simulate convection terms, while most models treat convection terms still only with traditional central differences. Response to direct wind forcing is satisfactory, but computed post wind events die out too quickly.

## Keywords

Numerical Diffusion Total Variation Diminishing Total Variation Diminishing Scheme Flux Correct Transport Artificial Diffusion## List of Symbols

**Roman****Symbols**- \(a(u) = \frac{\partial f(u)}{\partial u}\)
Characteristic convective speed

- \({\mathcal A}_{j+1/2}^{n}\)
Corrected (grid) anti-diffusive flux at the \(n\)-th time step

- \({\mathcal D}_{j+1/2}\)
Temporal mean value of the flux \(\varGamma \frac{\partial u}{\partial x}\): \({\mathcal D}_{j+1/2} = \frac{1}{\varDelta t}\int _{t^{n}}^{t^{n+1}}\left( \varGamma \frac{\partial u}{\partial x}\right) (x_{j+1/2}, t)\text{ d }t\)

- \(f(u)\)
Flux of \(u\)

- \({\mathcal F}_{j+1/2}\)
Temporal mean value of \(f\): \({\mathcal F}_{j+1/2} = \frac{1}{\varDelta x}\int _{t^{n}}^{t^{n+1}}f(x_{j+1/2},t)\text{ d }t\)

- \(g(x)\)
Initial distribution of a Heaviside-type function in a travelling shock wave, see (25.54)

- \({\textit{Pe}} = \frac{a\varDelta x}{\varGamma }\)
Grid-Péclet number, or cell Reynolds number

- \(\text{ sgn }(x)\)
\(\text{ sgn }(x) = \left\{ \begin{array}{ll}1,&{} x>0\\ \in [-1,1], &{} x= 0\\ -1, &{} x<0\end{array}\right. \)

- \(S_{j+1}^{n} \)
\(S_{j+1}^{n} = {\mathrm {sgn}}(U_{j+1}^{n} - U_{j}^{n})\)

- \(t\)
time variable

- \(T(z,0)\)
Initial vertical temperature distribution

- \(u\)
Differentiable function satisfying a conservation law

- \(U_{j}^{n}\)
Spatial mean value of \(u\) in grid point \(j\): \(U_{j}^{n} = \frac{1}{\varDelta x}\int _{x_{j-1/2}}^{x_{j+1/2}}u(x, t^{n})\text{ d }x\)

- \(U_{j+1/2}^{R,L}\)
Value of the linearly constructed \(u(x)\) at the right (\(R\)) or left (\(L\)) boundary of the grid with midpoint \(x_{j+1/2}\)

- \(x\)
Position

- \( x_{j}^{n}, f_{j}^{n}\)
Position, respectively function value at grid point \(j\) and at time slice \(n\).

**Greek****Symbols**- \(\varGamma \)
Turbulent diffusion coefficient

- \(\varGamma _{{\mathrm {num}}} = \frac{|a|\varDelta x}{2}\)
Numerical grid diffusivity

- \(\tilde{\varGamma }\)
Artificial diffusivity

- \(\varDelta t\)
Time step

- \(\delta x\)
Grid size

- \(\epsilon \)
Small positive increment of a variable

- \(\phi _j\)
Slope limiter \(\phi _j =\phi (\theta _j), \theta _{j} = (U_{j}- U_{j-1})/(U_{j+1} - U_{j})\)

- \(\phi ^{\mathrm {Superbee}}(\theta ) \)
\(\phi ^{\mathrm {Superbee}}(\theta ) = \max [0, \min (1, 2\theta ), \min (\theta ,2)\)

- \(\phi ^{\mathrm {Minmod}}(\theta )\)
\(\phi ^{\mathrm {Minmod}}(\theta ) = \max [0, \min (1,\theta )\)

- \(\phi ^{\mathrm {Woodward}}(\theta ) \)
\(\phi ^{\mathrm {Woodward}}(\theta ) = \max [0, \min (2, 2\theta , 0.5(1+\theta ))\)

- \(\sigma _{j} \)
slope limiter \(\sigma _{j} = \frac{1}{\varDelta x}\phi _{j}(U_{j+1} - U_{j})\)

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