Use of variational method to evaluate well-posedness of an advection-diffusion reaction equation of solute transport

Mwakilama, Elias; Gathungu, Duncan; Magagula, Vusi Mpendulo

doi:10.12988/ams.2022.916797

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“…Hence, to ensure a generally stable MV-SLQLM, two complementary measures are employed on the basis of solute transport equation, equation (13). First, in reference [86], we investigated the existence and uniqueness of the classical solution to equation (13) using the proposed "VF-EM" technique. Defining an error ẽ ≔ cðtÞ − cðtÞ and making inference of Ilati and Dehghan [82], the result (see equation 3.10 in [86]) means ẽ is strongly stable.…”

Section: Stabilitymentioning

confidence: 99%

“…First, in reference [86], we investigated the existence and uniqueness of the classical solution to equation (13) using the proposed "VF-EM" technique. Defining an error ẽ ≔ cðtÞ − cðtÞ and making inference of Ilati and Dehghan [82], the result (see equation 3.10 in [86]) means ẽ is strongly stable. Moreover, the well-posed analysis of equation (13) in time integral also demonstrates that the numerical technique is stable in L 2 -norm.…”

Section: Stabilitymentioning

confidence: 99%

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A Novel Multivariate Spectral Local Quasilinearization Method (MV-SLQLM) for Modelling Flow, Moisture, Heat, and Solute Transport in Soil

Mwakilama

Magagula

Gathungu

2023

Journal of Applied Mathematics

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Conventionally, the problem of studying the transport of water, heat, and solute in soil or groundwater systems has been numerically solved using finite difference (FD) or finite element (FE) methods. FE methods are attractive over FD methods because they are geometrically flexible. However, recent studies demonstrate that spectral collocation (SC) methods converge exponentially faster than FD or FE methods using a few grid points or on coarse grids. This work proposes and applies a multivariate spectral local quasilinearization method (MV-SLQLM) to model the transportation and interaction of soil moisture, heat, and solute concentration in a nonbare soil ridge. The MV-SLQLM uses a quasilinearization method (QLM) to linearize any nonlinear equations and then employs a local linearization method (LLM) to decouple the linearized system of PDEs to form a sequence of equations that are solved in a computationally efficient manner. The MV-SLQLM is thus an extension of the bivariate spectral local linearization method (BI-SLLM) that fails to deal with a 2D problem and is a modification of the MV-SQLM whose efficiency is compromised when operating on high dense solution matrices. We use the residual error norms of the difference between successive iterations to affirm convergence to the expected solution. To illustrate the application and check the solution accuracy, we conduct systematic analyses of the effect of model parameters on distribution profiles. Findings are in good agreement with theory and literature, thereby revealing suitability of the MV-SLQLM to solve coupled nonlinear PDEs with environmental fluid dynamics applications.

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