New vector norms, seminorms and exact solutions as a benchmark for the steady-state convection-diffusion equation

ALVAREZ, GUSTAVO B.; FONSECA, JÉSSICA M. DA; SOUSA, PATRÍCIA A.P. DE; STAMPA, CLEYTON S.

doi:10.1590/0001-3765202520241417

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ENGINEERING SCIENCES • An. Acad. Bras. Ciênc. 97 (4) • 2025 • https://doi.org/10.1590/0001-3765202520241417 linkcopy

New vector norms, seminorms and exact solutions as a benchmark for the steady-state convection-diffusion equation

Authorship SCIMAGO INSTITUTIONS RANKINGS

The steady-state convection-diffusion equation is part of the mathematical model for transport phenomena. Its exact solution may present boundary layers when convection is dominant. In these cases, numerical solutions of the second-order centered finite difference present spurious oscillations, and stabilized methods may present the smearing effect. Here new $h_{1}$ and $h_{2}$ vector norms and seminorms are proposed as analogs to the $H^{1}$ and $H^{2}$ norms used in the finite element framework, which allow defining new errors for the solutions and their derivatives obtained by the finite difference method. In addition, new $w l_{2}$ , $w h_{1}$ and $w h_{2}$ weighted norms and seminorms are introduced, allowing to observe convergences of the schemes similar to what should be expected theoretically. These norms and seminorms and their weighted versions are valid for uniform and non-uniform meshes. Furthermore, exact solutions with boundary layers are proposed as benchmarks together with the errors in the new vector norms of two classical finite difference schemes: centered and upwind. Numerical results indicate that a mesh that guarantees an acceptable approximation for the solution $u$ does not guarantee an acceptable approximation for the derivatives of $u$ . For this reason non-uniform meshes and new schemes will be analyzed in future articles.

Key words
diffusion - advection; finite difference method; vector norms; vector seminorms; exact solutions; stabilization

$h \to P_{e}$	${‖ E^{CC} ‖}_{l 2}$	${‖ E^{CC} ‖}_{h 1}$	${‖ E^{CC} ‖}_{h 2}$	${‖ E^{CC} ‖}_{w l 2}$	${‖ E^{CC} ‖}_{w h 1}$	${‖ E^{GU} ‖}_{w h 2}$
$h \to P_{e}$	${‖ E^{UPW} ‖}_{l 2}$	${‖ E^{UPW} ‖}_{h 1}$	${‖ E^{UPW} ‖}_{h 2}$	${‖ E^{UPW} ‖}_{w l 2}$	${‖ E^{UPW} ‖}_{w h 1}$	${‖ E^{UPW} ‖}_{w h 2}$
Problem with smooth solution: D = 0.01, v = 1.
0.1 → 5.0	3.64085×10⁻² 5.68378×10⁻¹	3.65066×10⁻² 6.31841×10⁻¹	2.69986×10⁻¹ 2.76062×10¹	4.04539×10⁻³ 6.31531×10⁻²	4.05629×10⁻³ 7.02046×10⁻²	2.99985×10⁻² 3.067361728
0.02 → 1.0	3.22451×10⁻³ 2.66262×10⁻¹	3.26954×10⁻³ 7.89313×10⁻¹	5.41713×10⁻² 7.43089×10¹	6.58064×10⁻⁵ 5.43392×10⁻³	6.67253×10⁻⁵ 1.61084×10⁻²	1.10553×10⁻³ 1.516509571
0.002 → 0.1	1.01923×10⁻⁴ 8.52480×10⁻²	1.06281×10⁻⁴ 7.94077×10⁻¹	3.01391×10⁻³ 7.89528×10¹	2.04256×10⁻⁷ 1.70837×10⁻⁴	2.12988×10⁻⁷ 1.59133×10⁻³	6.03990×10⁻⁶ 1.58222×10⁻²
Problem with smooth solution: D = 0.001, v = 1.
0.01 → 5.0	1.16094×10⁻³ 1.91273×10⁻¹	1.16112×10⁻³ 3.23473×10⁻¹	2.01289×10⁻² 2.60863×10²	1.17267×10⁻⁵ 1.93205×10⁻³	1.17284×10⁻⁵ 3.26741×10⁻³	2.03322×10⁻⁴ 2.634984716
0.002 → 1.0	1.03826×10⁻⁴ 8.58713×10⁻²	1.03956×10⁻⁴ 7.45467×10⁻¹	5.19409×10⁻³ 7.40505×10²	2.08069×10⁻⁷ 1.72086×10⁻⁴	2.08329×10⁻⁷ 1.49392×10⁻³	1.04090×10⁻⁵ 1.483978727
0.0002 → 0.1	3.28327×10⁻⁶ 2.71845×10⁻²	3.29673×10⁻⁶ 7.89824×10⁻¹	2.97586×10⁻⁴ 7.89356×10²	6.56786×10⁻¹⁰ 5.43798×10⁻⁶	6.59478×10⁻¹⁰ 1.57996×10⁻⁴	5.95291×10⁻⁸ 1.57902×10⁻¹
Problem with external boundary layer: D = 0.01, v = 1.
0.1 → 5.0	9.19527×10⁻¹ 9.12418×10⁻²	3.897894044 8.30389×10⁻¹	3.78808×10² 8.25403×10¹	1.02169×10⁻¹ 1.01379×10⁻²	4.33099×10⁻¹ 9.22655×10⁻²	4.20898×10¹ 9.171148054
0.02 → 1.0	1.36591×10⁻¹ 2.21746×10⁻¹	1.16153×10¹ 3.249384266	1.16151×10³ 3.24197×10²	2.78759×10⁻³ 4.52543×10⁻³	2.37047×10⁻¹ 6.63139×10⁻²	2.37042×10¹ 6.616269419
0.002 → 0.1	3.73969×10⁻³ 1.05867×10⁻¹	1.172428913 1.80439×10¹	1.17248×10² 1.80445×10³	7.49438×10⁻⁶ 2.12158×10⁻⁴	2.34955×10⁻³ 3.61603×10⁻²	2.34966×10⁻¹ 3.616149726
Problem with external boundary layer: D = 0.001, v = 1.
0.01 → 5.0	8.94461×10⁻¹ 9.12418×10⁻²	3.72447×10¹ 8.254119849	3.72339×10⁴ 8.25361×10³	9.03495×10⁻³ 9.21635×10⁻⁴	3.76209×10⁻¹ 8.33749×10⁻²	3.761009×10² 8.33698×10¹
0.002 → 1.0	1.36591×10⁻¹ 2.21746×10⁻¹	1.16145×10² 3.24188×10¹	1.16145×10⁵ 3.24181×10⁴	2.73731×10⁻⁴ 4.44381×10⁻⁴	2.32756×10⁻¹ 6.49676×10⁻²	2.32756×10² 6.49661×10¹
0.0002 → 0.1	3.73969×10⁻³ 1.05867×10⁻¹	1.17242×10¹ 1.80436×10²	1.17242×10⁴ 1.80436×10⁵	7.48089×10⁻⁷ 2.11776×10⁻⁵	2.34531×10⁻³ 3.60945×10⁻²	2.345316133 3.60946×10¹
Problem with internal boundary layer: D = 0.01, v = 1.
0.1 → 5.0	5.588840510 3.554068790	1.43144×10¹ 4.861175790	1.31791×10³ 3.31692×10²	6.20982×10⁻¹ 3.94896×10⁻¹	1.590499860 5.40131×10⁻¹	1.46435×10² 3.68547×10¹
0.02 → 1.0	9.66171×10⁻² 2.52825×10⁻¹	4.661116234 5.330358943	4.66034×10² 5.32462×10²	1.97177×10⁻³ 5.15969×10⁻³	9.51248×10⁻² 1.08782×10⁻¹	9.510914021 1.08665×10¹
0.002 → 0.1	2.34414×10⁻³ 8.38739×10⁻²	8.26431×10⁻² 2.309430925	8.261404872 2.30802×10²	4.69768×10⁻⁶ 1.68083×10⁻⁴	1.65617×10⁻⁴ 4.62811×10⁻³	1.65559×10⁻² 4.62529×10⁻¹
Problem with internal boundary layer: D = 0.1, v = 5.
0.2 → 5.0	1.339005717 1.161718403	1.819341166 1.281276560	6.16114×10¹ 2.70525×10¹	3.34751×10⁻¹ 2.90429×10⁻¹	4.54835×10⁻¹ 3.20319×10⁻¹	1.54028×10¹ 6.763142791
0.04 → 1.0	9.09779×10⁻² 2.35363×10⁻¹	7.72443×10⁻¹ 3.050616716	3.83611×10¹ 1.52106×10²	3.79074×10⁻³ 9.80682×10⁻³	3.21851×10⁻² 1.27109×10⁻¹	1.598381182 6.337782408
0.004 → 0.1	2.34414×10⁻³ 8.38738×10⁻²	4.13714×10⁻² 1.156997805	2.065662213 5.77092×10¹	9.41423×10⁻⁶ 3.36842×10⁻⁴	1.66150×10⁻⁴ 4.64657×10⁻³	8.29583×10⁻³ 2.31764×10⁻¹

h, $p_{e}$	$‖ E^{CC} ‖ l_{2}$ , $‖ E^{UPW} ‖ l_{2}$	$‖ E^{CC} ‖ h_{1}$ , $‖ E^{UPW} ‖ h_{1}$	$‖ E^{CC} ‖ h_{2}$ , $‖ E^{UPW} ‖ h_{2}$	$‖ E^{CC} ‖ {w l}_{2}$ , $‖ E^{UPW} ‖ {w l}_{2}$	$‖ E^{CC} ‖ {w h}_{1}$ , $‖ E^{UPW} ‖ {w h}_{1}$	$‖ E^{GU} ‖ {w h}_{2}$ , $‖ E^{UPW} ‖ {w h}_{2}$
h, $p_{e}$	Problem with smooth solution: $D = 0.02, v = (1, 2)$
0.018181818 / 1.016394535	1.07455×10⁻² / 1.18959	4.03918×10⁻² / 5.28765	2.21253 / 3.79997×10²	3.42653×10⁻⁶ / 3.79333×10⁻⁴	1.28800×10⁻⁵ / 1.68611×10⁻³	7.05526×10⁻⁴ / 1.21172×10⁻¹
0.0018018018 / 0.100723782	1.06476×10⁻³ / 1.22746	5.27423×10⁻³ / 1.30370×10¹	3.77710×10⁻¹ / 1.16313×10³	3.44432×10⁻⁹ / 3.97061×10⁻⁶	1.70612×10⁻⁸ / 4.21724×10⁻⁵	1.22182×10⁻⁶ / 3.76252×10⁻³
Problem with smooth solution: $D = 0.002, v = (1, 2)$
0.001 / 0.559016994	6.42071×10⁻⁴ / 1.29028	7.53284×10⁻³ / 2.26437×10¹	6.26101 / 1.96258×10⁴	6.40789×10⁻¹⁰ / 1.28769×10⁻⁶	7.51779×10⁻⁹ / 2.25985×10⁻⁵	6.24851×10⁻⁶ / 1.95866×10⁻²
Problem with external boundary layer: $D = 0.02, v = (1, 2)$
0.0018018018 / 0.100723782	7.79735×10⁻² / 2.8550	2.39500×10¹ / 4.42838×10²	2.36740×10³ / 4.17801×10⁴	2.52230×10⁻⁷ / 9.23551×10⁻⁶	7.74741×10⁻⁵ / 1.43250×10⁻³	7.65814×10⁻³ / 1.35151×10⁻¹
Problem with internal boundary layer: $D = 0.2, v = (5, 10)$
0.003571428 / 0.099824463	2.76336×10⁻² / 1.14993	8.25561×10⁻¹ / 2.15842×10¹	5.67455×10¹ / 1.09510×10³	3.49965×10⁻⁷ / 1.45633×10⁻⁵	1.04553×10⁻⁵ / 2.73353×10⁻⁴	7.18652×10⁻⁴ / 1.38688×10⁻²

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