17In this paper, we develop a finite analytic method (FAMM), which combines 18 flexibility of numerical methods and advantages of analytical solutions, to solve the 19 mixed-form Richards' equation. This new approach minimizes mass balance errors 20 and truncation errors associated with most numerical approaches. We use numerical 21 experiments to demonstrate that FAMM can obtain more accurate numerical solutions 22 32 33 Keywords: Mixed-form Richards' equation, Finite analytic method, Analytical 34 solution, Mass conservative property 35 36 Abbreviations: 37 FAMM, finite analytic method based on mixed-form Richards' equation; 38 MPFD, modified Picard finite difference approximation; 39 FAMH, finite analytic method based on head-based Richards' equation.40 3 55 2011, Yeh et al., 2015a). However, Milly (1985) and Celia et al. (1990) reported that 56 the h-based Richards' equation is difficult to solve using numerical approaches. 57 Numerical solutions to this equation may yield results involving large mass balance 58 errors, unless small grids and fine time steps are employed. In order to avoid these 59 problems, many researchers have suggested that the mixed-form Richards' equation, 60 which can maintain the conservative property with less computational efforts, should 61 be adopted to simulate water flow in unsaturated zone (Allen and Murphy, 1986; 62 4 Celia et al., 1987; Celia et al., 1990). 63 Celia et al. (1990) developed a modified Picard finite difference (MPFD) method 64 for the mixed-form Richards' equation and showed that it yielded robust and reliable 65 numerical solutions for unsaturated flow problems. They stated that the mixed-form 66 equation combines advantages in the h -based and the  -based equations while 67 circumventing difficulties associated with each one. Their numerical solutions to the 68 mixed-form Richards' equation, nevertheless, are subject to truncation error as the 69 most numerical approaches. 70 A finite analytic method (FAM) was presented by Chen and Chen (1981, 1984) to 71 solve heat conduction, and Navier-Strokes equations. Zeng and Li(1987) advocated 72 that the finite analytic method could minimize the truncation errors and yield stable 73 numerical solutions. Hwang et al. (1985) applied FAM to solving two-dimensional 74 solute transport equation, and reported that FAM produced accurate results and 75 eliminated numerical dispersion for large Peclet numbers. A hybrid Laplace 76 transform finite analytic method (LTFAM) was developed by Wang et al. (2012) for 77 solving advection-dispersion equations. Comparing results of LTFAM with those of 78 the analytical solutions, they concluded that the LTFAM generated highly accurate 79 numerical solutions even under the conditions where Peclet numbers are greater than 80 50. Using an optimal time-weighting factor, Tsai et al. (1993) built an FAM for 81 solving the h-based Richards' equation with irregular boundaries. By combining 82 with fine-point local elements and nine-point local elements, they reported that the 83 FAM could ...

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The unsaturated zone plays an important role in groundwater recharge, discharge, and the ecological environment. Mathematical models are essential to the investigations of low processes in the unsaturated zone. The governing equations of the mathematic models for unsaturated low are nonlinear since their unsaturated hydraulic parameters of the equations depend on their solutions (i.e., pressure or moisture content). The nonlinearity can become very strong with low in relatively dry soils, and the equation has hyperbolic characteristics. Therefore, the traditional numerical methods, such as the inite difference method and the inite element method, could lead to the numerical oscillation, dispersion, and the divergence of the solution if improper time and space steps were selected. In this study, a inite analytic method (FAM) to solve Richards' equation was developed. The basic idea of the FAM is the incorporation of the analytic solution in a small local element to formulate the algebraic representation of the partial differential equation of the unsaturated low. Therefore, the inite analytic numerical method can effectively control the numerical oscillation and dispersion. The convergence and stability of inite analytic numerical scheme are then proven by a rigorous mathematical analysis. Numerical experiments are then used to show that FAM is highly accurate by comparing its results with those obtained using analytical solutions by a modiied Picard inite difference method. In addition, we also demonstrate that FAM reproduces results of laboratory experiments. Therefore, it can be considered an appropriate simulation method for the unsaturated low.Abbreviations: FAM, inite analytic method; LTFAM, Laplace transform inite analytic method; MPFD, modiied Picard inite difference approximation.

Ground soil heat flux, G 0 , is a difficult-to-measure but important component of the surface energy budget. Over the past years, many methods were proposed to estimate G 0 ; however, the application of these methods was seldom validated and assessed under different weather conditions. In this study, three popular models (force-restore, conduction-convection, and harmonic) and one widely used method (plate calorimetric), which had well performance in publications, were investigated using field data to estimate daily G 0 on clear, cloudy, and rainy days, while the gradient calorimetric method was regarded as the reference for assessing the accuracy. The results showed that harmonic model was well reproducing the G 0 curve for clear days, but it yielded large errors on cloudy and rainy days. The force-restore model worked well only under rainfall condition, but it was poor to estimate G 0 under rain-free conditions. On the contrary, the conduction-convection model was acceptable to determine G 0 under rain-free conditions, but it generated large errors on rainfall days. More importantly, the plate calorimetric method was the best to estimate G 0 under different weather conditions compared with the three models, but the performance of this method is affected by the placement depth of the heat flux plate. As a result, the heat flux plate was recommended to be buried as close as possible to the surface under clear condition. But under cloudy and rainy conditions, the plate placed at depth of around 0.075 m yielded G 0 well. Overall, the findings of this paper provide guidelines to acquire more accurate estimation of G 0 under different weather conditions, which could improve the surface energy balance in field.