Research Article
Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians
Jared Stewart*,
Mayya Tokman,
Valentin Dallerit,
Fabrizio Bisetti,
Oscar Diaz-Ibarra
Issue:
Volume 13, Issue 2, April 2024
Pages:
29-37
Received:
18 February 2024
Accepted:
19 March 2024
Published:
7 April 2024
Abstract: Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.
Abstract: Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate co...
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Research Article
A Reaction-diffusion System Modeling the Transmission of Typhoid Fever in a Periodic Environment
Huei-Li Lin,
Yu-Chiau Shyu,
Chih-Lang Lin,
Feng-Bin Wang
Issue:
Volume 13, Issue 2, April 2024
Pages:
38-52
Received:
13 March 2024
Accepted:
27 March 2024
Published:
21 April 2024
Abstract: Typhoid fever is a life-threatening infection caused by the bacterium Salmonella Typhi, and it is still an important issue in developing countries. There are two infection routes of Typhoid fever, namely, the human-to-human transmission and the environment-to-human transmission. It is evident that people living near rivers may have a higher rate of typhoid infection, and temperature changes also have significant impacts on Typhoid transmission dynamics. In the model, the population of human will be divided into susceptible individuals, infected individuals, carrier individuals, individuals under treatment, and recovered individuals. Then a periodic dispersion-reaction system is used to describe the transport and the interactions between human and bacteria in the environment. The solution maps of the proposed periodic dispersion-reaction system lack the compactness since the population under treatment has no diffusion term, which makes analysis more difficult. After the feasible domain is chosen carefully, the eventually boundedness of the solutions can be established, and the loss of compactness is overcome if the initial data is chosen from the feasible domain. In order to introduce the reproduction number R0, the linearized system around the disease-free state is constructed, and the basic reproduction number is defined as the spectral radius of the next generation operator. Then the comparison principle and persistence theory can be utilized to establish that the index R0 completely determines the threshold behavior of the typhoid spread. Brief mathematical and biological interpretations are also presented.
Abstract: Typhoid fever is a life-threatening infection caused by the bacterium Salmonella Typhi, and it is still an important issue in developing countries. There are two infection routes of Typhoid fever, namely, the human-to-human transmission and the environment-to-human transmission. It is evident that people living near rivers may have a higher rate of...
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