A Finite Difference Scheme for the Richards Equation Under Variable-Flux Boundary

2021 ◽  
pp. 231-245
Author(s):  
Yasuhide Fukumoto ◽  
Fengnan Liu ◽  
Xiaopeng Zhao
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Richards Equation ◽  
Variable Flux

scholarly journals Kalman filters for assimilating near-surface observations in the Richards equation – Part 1: Retrieving state profiles with linear and nonlinear numerical schemes

2012 ◽  
Vol 9 (12) ◽  
pp. 13291-13327 ◽  
Author(s):  
G. B. Chirico ◽  
H. Medina ◽  
N. Romano
Keyword(s):  
Kalman Filter ◽  
Finite Difference ◽  
Difference Scheme ◽  
Soil Water ◽  
Finite Difference Scheme ◽  
Richards Equation ◽  
Near Surface ◽  
Backward Euler ◽  
Explicit Finite Difference ◽  
Surface Observations

Abstract. This paper examines the potential of different algorithms, based on the Kalman filtering approach, for assimilating near-surface observations in a one-dimensional Richards' equation. Our specific objectives are: (i) to compare the efficiency of different Kalman filter algorithms, implemented with different numerical schemes of the Richards equation, in retrieving soil water potential profiles; (ii) to evaluate the performance of these algorithms when nonlinearities arise from the nonlinearity of the observation equation, i.e. when surface soil water content observations are assimilated to retrieve pressure head values. The study is based on a synthetic simulation of an evaporation process from a homogeneous soil column. A standard Kalman Filter algorithm is implemented with both an explicit finite difference scheme and a Crank-Nicolson finite difference scheme of the Richards equation. Extended and Unscented Kalman Filters are instead both evaluated to deal with the nonlinearity of a backward Euler finite difference scheme. While an explicit finite difference scheme is computationally too inefficient to be implemented in an operational assimilation scheme, the retrieving algorithm implemented with a Crank-Nicolson scheme is found computationally more feasible and robust than those implemented with the backward Euler scheme. The Unscented Kalman Filter reveals as the most practical approach when one has to deal with further nonlinearities arising from the observation equation, as result of the nonlinearity of the soil water retention function.

scholarly journals Kalman filters for assimilating near-surface observations into the Richards equation – Part 1: Retrieving state profiles with linear and nonlinear numerical schemes

2014 ◽  
Vol 18 (7) ◽  
pp. 2503-2520 ◽  
Author(s):  
G. B. Chirico ◽  
H. Medina ◽  
N. Romano
Keyword(s):  
Kalman Filter ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Pressure Head ◽  
Observation Equation ◽  
Richards Equation ◽  
Numerical Schemes ◽  
Near Surface ◽  
Backward Euler

Abstract. This paper examines the potential of different algorithms, based on the Kalman filtering approach, for assimilating near-surface observations into a one-dimensional Richards equation governing soil water flow in soil. Our specific objectives are: (i) to compare the efficiency of different Kalman filter algorithms in retrieving matric pressure head profiles when they are implemented with different numerical schemes of the Richards equation; (ii) to evaluate the performance of these algorithms when nonlinearities arise from the nonlinearity of the observation equation, i.e. when surface soil water content observations are assimilated to retrieve matric pressure head values. The study is based on a synthetic simulation of an evaporation process from a homogeneous soil column. Our first objective is achieved by implementing a Standard Kalman Filter (SKF) algorithm with both an explicit finite difference scheme (EX) and a Crank-Nicolson (CN) linear finite difference scheme of the Richards equation. The Unscented (UKF) and Ensemble Kalman Filters (EnKF) are applied to handle the nonlinearity of a backward Euler finite difference scheme. To accomplish the second objective, an analogous framework is applied, with the exception of replacing SKF with the Extended Kalman Filter (EKF) in combination with a CN numerical scheme, so as to handle the nonlinearity of the observation equation. While the EX scheme is computationally too inefficient to be implemented in an operational assimilation scheme, the retrieval algorithm implemented with a CN scheme is found to be computationally more feasible and accurate than those implemented with the backward Euler scheme, at least for the examined one-dimensional problem. The UKF appears to be as feasible as the EnKF when one has to handle nonlinear numerical schemes or additional nonlinearities arising from the observation equation, at least for systems of small dimensionality as the one examined in this study.

scholarly journals The convergence of a numerical method for total variation flow

2021 ◽  
Vol 15 ◽  
pp. 174830262110113
Author(s):  
Qianying Hong ◽  
Ming-jun Lai ◽  
Jingyue Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Total Variation ◽  
Iterative Algorithm ◽  
Finite Difference Scheme ◽  
Gradient Flow ◽  
Time Dependent ◽  
Total Variation Flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

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