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A Local Discontinuous Galerkin Dual-Time Richards' Equation Solution and Analysis on Dual-Time Stability and Convergence

Xiao, Yilong
http://orcid.org/0000-0001-5024-1652
http://rave.ohiolink.edu/etdc/view?acc_num=osu1620696581506058

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Civil Engineering.
Richards' equation (RE) governs single-phase, variably saturated flows driven by gravity and pressure in porous media and can be used to simulate soil-water flow and associated processes. Solving RE is complicated by factors such as highly nonlinear relationships among soil hydraulic parameters, heterogeneity in soil properties, and the presence of sharp wetting fronts during infiltration into dry soils. These factors lead to two major active challenges: (i) a lack of higher-order accuracy in space, and (ii) a lack of robustness and convergence in implicit time steppers especially under extreme conditions. We tackle these challenges through developing a high-order RE solver based on a local discontinuous Galerkin finite-element method (LDG-FEM or LDG) in space and a dual-time (DT) stepping method in time. LDG achieves high-order accuracy via the use of high-degree basis polynomials, whereas DT resolves lack of convergence in implicit time steppers by turning the original transient problem into a steady-state problem. Our solver is successfully verified against several analytical solutions, showing more resilience and higher accuracy than direct application of the implicit method in the transient form of RE. However, despite the success of DT methods in literature, some of their fundamental aspects such as stability and convergence criteria are inadequately addressed. Often only provided in research articles are recommended ratios of DT step sizes for stability. No conclusion has been reached on how DT convergence rate can be optimized in general, nor is there concrete supportive evidence for the use of high-order pseudo-time schemes given that accuracy is determined by the physical-time scheme. We conducted a stability and convergence analysis to address these aspects for DT methods which couple second-order backward differential formula in physical-time with a fully explicit residual. It is discovered that the ratio of pseudo- to physical-time steps directly influences convergence rate. For linear problems, a closed-formed equation is derived to return the optimal ratio for convergence based on physical parameters.
Ethan Kubatko, Ph.D. (Advisor)
Gil Bohrer, Ph.D. (Committee Member)
Edward McCoy, Ph.D. (Committee Member)
Daniel Pradel, Ph.D. (Committee Member)
113 p.
Civil Engineering; Environmental Engineering; Geotechnology; Hydrology
Richards Equation; Soil-Water Flow; Dual-Time Stepping Method; Pseudo-Time Stepping Method; Convergence

Recommended Citations

Citations

  • Xiao, Y. (2021). A Local Discontinuous Galerkin Dual-Time Richards' Equation Solution and Analysis on Dual-Time Stability and Convergence [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1620696581506058

    APA Style (7th edition)

  • Xiao, Yilong. A Local Discontinuous Galerkin Dual-Time Richards' Equation Solution and Analysis on Dual-Time Stability and Convergence. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1620696581506058.

    MLA Style (8th edition)

  • Xiao, Yilong. "A Local Discontinuous Galerkin Dual-Time Richards' Equation Solution and Analysis on Dual-Time Stability and Convergence." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1620696581506058

    Chicago Manual of Style (17th edition)

osu1620696581506058
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This open access ETD is published by The Ohio State University and OhioLINK.