Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems

Vincenzo Angelo Pennati; Antoine Celestin Kengni Jotsa; Jacques Tagoudjeu

doi:doi:10.11648/j.pamj.20251404.11

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Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems

Vincenzo Angelo Pennati^*

, Antoine Celestin Kengni Jotsa

, Jacques Tagoudjeu

Published in Pure and Applied Mathematics Journal (Volume 14, Issue 4)

Received: 8 June 2025 Accepted: 30 June 2025 Published: 15 August 2025

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Abstract

Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.

Published in	Pure and Applied Mathematics Journal (Volume 14, Issue 4)
DOI	10.11648/j.pamj.20251404.11
Page(s)	69-92
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

FVM for Convective-diffusive Problems, High Order Monotonic Schemes, Convergence for Initial-boundary Values Problems, Dynamically Local Self h-Adaptivity

References

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Pennati, V. A., Jotsa, A. C. K., Tagoudjeu, J. (2025). Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems. Pure and Applied Mathematics Journal, 14(4), 69-92. https://doi.org/10.11648/j.pamj.20251404.11

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ACS Style

Pennati, V. A.; Jotsa, A. C. K.; Tagoudjeu, J. Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems. Pure Appl. Math. J. 2025, 14(4), 69-92. doi: 10.11648/j.pamj.20251404.11

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AMA Style

Pennati VA, Jotsa ACK, Tagoudjeu J. Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems. Pure Appl Math J. 2025;14(4):69-92. doi: 10.11648/j.pamj.20251404.11

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@article{10.11648/j.pamj.20251404.11,
  author = {Vincenzo Angelo Pennati and Antoine Celestin Kengni Jotsa and Jacques Tagoudjeu},
  title = {Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {4},
  pages = {69-92},
  doi = {10.11648/j.pamj.20251404.11},
  url = {https://doi.org/10.11648/j.pamj.20251404.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251404.11},
  abstract = {Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems

AU  - Vincenzo Angelo Pennati
AU  - Antoine Celestin Kengni Jotsa
AU  - Jacques Tagoudjeu
Y1  - 2025/08/15
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251404.11
DO  - 10.11648/j.pamj.20251404.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 69
EP  - 92
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251404.11
AB  - Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.
VL  - 14
IS  - 4
ER  -

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Vincenzo Angelo Pennati

Theoretical Analysis Service, Hydraulic & Structural Research Center, ENEL, Milano, Italy

Contact Email

http://orcid.org/0000-0002-0770-7645
Antoine Celestin Kengni Jotsa

Department of Fundamental Sciences, National Advanced School of Mines and Petroleum Industries at Kaele, University of Maroua, Maroua, Cameroon

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http://orcid.org/0000-0003-3395-2561
Jacques Tagoudjeu

Department of Mathematics and Physical Sciences, National Advanced School of Engineering of Yaounde, University of Yaounde I, Yaounde, Cameroon

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http://orcid.org/0000-0002-9379-3171

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Pennati, V. A., Jotsa, A. C. K., Tagoudjeu, J. (2025). Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems. Pure and Applied Mathematics Journal, 14(4), 69-92. https://doi.org/10.11648/j.pamj.20251404.11

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ACS Style

Pennati, V. A.; Jotsa, A. C. K.; Tagoudjeu, J. Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems. Pure Appl. Math. J. 2025, 14(4), 69-92. doi: 10.11648/j.pamj.20251404.11

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AMA Style

Pennati VA, Jotsa ACK, Tagoudjeu J. Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems. Pure Appl Math J. 2025;14(4):69-92. doi: 10.11648/j.pamj.20251404.11

Copy | Download

@article{10.11648/j.pamj.20251404.11,
  author = {Vincenzo Angelo Pennati and Antoine Celestin Kengni Jotsa and Jacques Tagoudjeu},
  title = {Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {4},
  pages = {69-92},
  doi = {10.11648/j.pamj.20251404.11},
  url = {https://doi.org/10.11648/j.pamj.20251404.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251404.11},
  abstract = {Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems

AU  - Vincenzo Angelo Pennati
AU  - Antoine Celestin Kengni Jotsa
AU  - Jacques Tagoudjeu
Y1  - 2025/08/15
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251404.11
DO  - 10.11648/j.pamj.20251404.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 69
EP  - 92
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251404.11
AB  - Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.
VL  - 14
IS  - 4
ER  -

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