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ABSTRACT
The spatial discretization of diffusion terms leads to a theoretical order loss, specially in the presence of strong property and variable gradients. New conservative and nondispersive numerical schemes are proposed to circumvent this issue. Their spectral resolution, numerical order, and cost are evaluated and compared to traditional finite volume and difference schemes. Cost is measured through the computer time and memory it takes these schemes to reach a user-prescribed error tolerance. A manufactured test case is constructed to evaluate these characteristics. Its analysis shows that the new schemes are more cost-effective than traditional schemes under the conditions analyzed.
Nomenclature
| a | = | smallest nonzero eigenvalue |
| Cp | = | heat capacity at constant pressure |
| e | = | diffusion flux |
| f | = | property |
| g | = | primitive variable |
| hl | = | convective heat-transfer coefficient |
| k | = | thermal conductivity |
| L | = | domain length |
| Lq | = | q-norm used to evaluate the steady temperature absolute error |
| N | = | number of mesh points |
| p | = | numerical order |
| q | = | manufactured source term |
| r | = | mesh ratio |
| t | = | time |
| T | = | temperature |
| T0 | = | initial temperature |
| TL | = | external fluid temperature |
| x | = | spatial coordinate |
| α | = | wave number |
| = | modified wave number | |
| γ | = | discrete operator |
| δ | = | discrete operator |
| Δx | = | mesh spacing |
| ϵ | = | discrete operator |
| ρ | = | density |
| θ | = | single parameter |
| Subscripts | = | |
| 0 | = | initial, left |
| i | = | mesh point |
| L | = | right |
| Superscripts | = | |
| m | = | accuracy order |
| n | = | accuracy order |
| o | = | accuracy order |
Nomenclature
| a | = | smallest nonzero eigenvalue |
| Cp | = | heat capacity at constant pressure |
| e | = | diffusion flux |
| f | = | property |
| g | = | primitive variable |
| hl | = | convective heat-transfer coefficient |
| k | = | thermal conductivity |
| L | = | domain length |
| Lq | = | q-norm used to evaluate the steady temperature absolute error |
| N | = | number of mesh points |
| p | = | numerical order |
| q | = | manufactured source term |
| r | = | mesh ratio |
| t | = | time |
| T | = | temperature |
| T0 | = | initial temperature |
| TL | = | external fluid temperature |
| x | = | spatial coordinate |
| α | = | wave number |
| = | modified wave number | |
| γ | = | discrete operator |
| δ | = | discrete operator |
| Δx | = | mesh spacing |
| ϵ | = | discrete operator |
| ρ | = | density |
| θ | = | single parameter |
| Subscripts | = | |
| 0 | = | initial, left |
| i | = | mesh point |
| L | = | right |
| Superscripts | = | |
| m | = | accuracy order |
| n | = | accuracy order |
| o | = | accuracy order |