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A steady forced Korteweg–de Vries (fKdV) model which includes gravity, capillary, and pressure distributions is solved numerically using the wavelet Galerkin method. The anti-derivatives of Daubechies wavelets are developed as the basis of the solution subspaces for the mixed boundary condition type. Accuracy of numerical solutions can be improved by increasing the number of wavelet levels in the multi-resolution analysis. The theoretical result of convergence rate is also shown. The problem can be viewed as gravity-capillary wave flows over an applied pressure distribution. The flow regime can be characterized by subcritical, supercritical, and critical flows depending on the value of the Froude number. Trapped depression and elevation waves are found over the pressure distribution. For a near-critical flow regime, a generalized solitary wave with ripples is presented. This shows a capillary effect in balance to gravity and the pressure force on the free surface.

A modified fully nonlinear model of an air–water system in deep water is presented in which the effect of wind in the air is simply represented by a direct link between the air–water interface pressure and the interface slope. The water system is a fully nonlinear Euler model of incompressible and irrotational fluid flow. Our main aim is to establish and compare with a reduced model represented by a forced nonlinear Schrödinger (FNLS) equation that describes wave groups in a weakly nonlinear asymptotic limit. Wave groups are described by the soliton and breather solutions generated by four cases of initial conditions in the relevant parameter regime. Numerical simulations of wave group formation in both models are compared, both with and without wind forcing. The FNLS model gives a good prediction to the modified fully nonlinear model when the wavenumber and wave frequency of the initial carrier waves are close to unity in dimensionless units based on typical carrier wavenumber and wave frequency. Wind forcing induces an exponential growth rate in the maximum amplitude wave. When the wave steepness becomes high in the fully nonlinear model some wave breaking is observed, but the FNLS model continues to predict large waves without breaking and there is then agreement only in the initial stage for the relevant initial conditions and parameter value ranges.

The rainfall data set usually contains missing values due to easily broken sensors. In Thailand, many public agencies collect rainfall values, including National Hydro Informatics (HII), Thai Meteorological Department, etc., since the data are valuable in terms of rainfall prediction, which is important for an agricultural country like Thailand. The rainfall data is normally collected hourly, and because there are many sensor locations, it is hard to maintain these sensors. The sensor data can be lost transiently and/or may yield anomaly values. Since there is a lot of data flowing to the server every day, it is hard to inspect manually or even semi-manually. This project collaborates with HII to develop a system that automates the rainfall data quality improvement process. The machine learning algorithms are used as tools for data cleansing. The derived data can be exposed as an open data set for many developers to explore new innovations. We explore data set characteristics and adopt both statistical and machine learning methods. The results show that the approach used both statistical and machine learning resulting in higher accuracy than using only statistical or machine learning approaches. We also develop a web application to visualize rainfall data results after cleansing and be connected to the models for the automatic cleansing pipelines.

This paper extends a previous study of free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation, to an analogous study of flow over two localised holes, or a combination of an obstacle and a hole. Importantly the terminology obstacle or hole can be reversed for a stratified fluid and refers more precisely to the relative polarity of the forcing and the solitary wave solution of the unforced Korteweg–de Vries equation. As in the previous study, our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. In the transcritical regime at early times, undular bores are produced upstream and downstream of each forcing site. We then describe the interaction of these undular bores between the forcing sites, and the outcome at very large times.

A time delay epidemic model is presented for the spread of the Coronavirus 2019 (COVID-19) in China. The time delay effects affect infected individuals. Monte Carlo simulation is performed to estimate the transmission and recovery rates. The basic reproduction number is estimated in terms of the average infected ratio. This ratio can be used to monitor the policy performance of disease control during the spread of the disease.

A forced extended Korteweg-de Vries equation is used to simulate the transcritical flow of a density stratified fluid over localized topography. The extension of the usual Korteweg-de Vries equation is by the addition of a cubic nonlinear term as well as the usual quadratic nonlinear term, which is motivated by certain density stratifications, such as a two-layer fluid model. The case of negative forcing corresponds to a near-surface sharp pycnocline over an obstacle.

Finite volume method with reconstruction and bottom modification techniques for simulating open channel flows in arbitrary cross-sectional areas is presented. These techniques are introduce...

Water level data obtained from telemetry stations typically contains large number of outliers. Anomaly detection and a data imputation are necessary steps in a data monitoring system. Anomaly data can be detected if its values lie outside of a normal pattern distribution. We developed a median-based statistical outlier detection approach using a sliding window technique. In order to fill anomalies, various interpolation techniques were considered. Our proposed framework exhibited promising results after evaluating with F1-score and root mean square error (RMSE) based on our artificially induced data points. The present system can also be easily applied to various patterns of hydrological time series with diverse choices of internal methods and fine-tuned parameters. Specifically, the Spline interpolation method yielded a superior performance on non-cyclical data while the long short-term memory (LSTM) outperformed other interpolation methods on a distinct tidal data pattern.

The multilevel augmentation method with the anti-derivatives of the Daubechies wavelets is presented for solving nonlinear two-point boundary value problems. The anti-derivatives of the Daubechies wavelets are applied as the multilevel bases for the subspaces of approximate solutions. This process results in a full nonlinear system that can be solved by the multilevel augmentation method for reducing computational cost. The convergence rate of the present method is shown. It is the order of $2^{s}$ , $0\leq s\leq p$ when p is the order of the Daubechies wavelets. Various examples of the Dirichlet boundary conditions are shown to confirm the theoretical results.

We consider free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation in a suite of numerical simulations. Our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. The flow behaviour can be characterised by the Froude number and the maximum heights of the obstacles. In the transcritical regime at early times, undular bores are produced upstream and downstream of each obstacle. Our main aim is to describe the interaction of these undular bores between the obstacles, and to find the outcome at very large times. We find that the flow development can be defined in three stages. The first stage is described by the well-known development of undular bores upstream and downstream of each obstacle. The second stage is the interaction between the undular bore moving downstream from the first obstacle and the undular bore moving upstream from the second obstacle. The third stage is the very large time evolution of this interaction, when one of the obstacles controls criticality. For equal obstacle heights, our analytical and numerical results indicate that either one of the obstacles can control flow criticality, that being the first obstacle when the flow is slightly subcritical and the second obstacle otherwise. For unequal obstacle heights the larger obstacle controls criticality. The results obtained here complement a recent numerical study using the fully nonlinear, but non-dispersive, shallow water equations.

A well-balanced finite volume method for solving two-dimensional shallow water equations with weighted average flux (WAF) is developed in this work to simulate flooding. Friction source terms are estimated with a semi-implicit scheme resulting in an efficient numerical method for simulating shallow water flows over irregular domains, for both wet and dry beds. A wet/dry cell tracking technique is also presented for reducing computational time. The accuracy of these methods are investigated by application to well-studied cases. For practical purposes, the developed scheme is applied to simulate the flooding of the Chao Phraya river from Chai Nat to Sing Buri provinces in Thailand during October 13–17, 2011. The numerical simulations yield results that agree with the existing data obtained from the satellite images.

The nonlinear shallow-water equations are often used to model flow over topography. In this paper we use these equations both analytically and numerically to study flow over two widely separated localised obstacles, and compare the outcome with the corresponding flow over a single localised obstacle. Initially we assume uniform flow with constant water depth, which is then perturbed by the obstacles. The upstream flow can be characterised as subcritical, supercritical and transcritical, respectively. We review the well-known theory for flow over a single localised obstacle, where in the transcritical regime the flow is characterised by a local hydraulic flow over the obstacle, contained between an elevation shock propagating upstream and a depression shock propagating downstream. Classical shock closure conditions are used to determine these shocks. Then we show that the same approach can be used to describe the flow over two widely spaced localised obstacles. The flow development can be characterised by two stages. The first stage is the generation of upstream elevation shock and downstream depression shock from each obstacle alone, isolated from the other obstacle. The second stage is the interaction of two shocks between the two obstacles, followed by an adjustment to a hydraulic flow over both obstacles, with criticality being controlled by the higher of the two obstacles, and by the second obstacle when they have equal heights. This hydraulic flow is terminated by an elevation shock propagating upstream of the first obstacle and a depression shock propagating downstream of the second obstacle. A weakly nonlinear model for sufficiently small obstacles is developed to describe this second stage. The theoretical results are compared with fully nonlinear simulations obtained using a well-balanced finite-volume method. The analytical results agree quite well with the nonlinear simulations for sufficiently small obstacles.

This paper developed the anti-derivative wavelet bases to handle the more general types of boundary conditions: Dirichlet, mixed and Neumann boundary conditions. The boundary value problem can be formulated by the variational approach, resulting in a system involving unknown wavelet coefficients. The wavelet bases are constructed to solve the unknown solutions corresponding to the types of solution spaces. The augmentation method is presented to reduce the dimension of the original system, while the convergence rate is in the same order as the multiresolution method. Some numerical examples have been shown to confirm the rate of convergence. The examples of the singularly perturbed problem with Neumann boundary conditions are also demonstrated, including highly oscillating cases.

Multilevel augmentation method with wavelet bases is demonstrated to show as the fast technique for solving singularly perturbed problems. Linear and quadratic wavelet bases are employed for constructing the full form of matrix system. To reduce the size of matrix coefficients, the multilevel augmented technique is applied at each current basis level. It is found that the multilevel augmentation method is faster than the standard multilevel method at the same order of accuracy. Convergent rates for linear and quadratic bases are 2 and 4 respectively. By the application of wavelet bases, numerical accuracy can be easily improved by increasing just desired levels in the multilevel augmentation process.

A mathematical model for DO (dissolved oxygen) and BOD (biochemical oxygen demand) in a non-uniform open channel is presented in this work. This model can be used to study transport processes in one-dimensional flow. An accurate numerical scheme based on the finite element method with linear polynomial basis is developed for studying flows both upstream and downstream with respect to a point source. The accuracy of numerical solutions have been studied and compared with certain analytical solutions at steady state. Various test cases including only dispersion effect, DO sag curve, and the linear variation of cross-sectional area are investigated. We have also derived an analytical solution for the last case when the model is simplified under certain conditions. This model has been applied to real flow parameters of Tha Chin river in Thailand. Furthermore, the impacts of flow velocity, kinetic reaction order, and dispersion coefficients have been simulated to observe the dissolved oxygen and biochemical oxygen demand dynamics along the channel.

This paper presents the approach to estimate the water height and its speed in a dam-break problem using the shallow water equations. The finite volume model is implemented using a CUDA program to run on a GPU. CUDA is a parallel programming model developed by NVIDIA. The results have been tested on the large scale problems, i.e., 48 × 48 and 240 × 240 cells. The results have been shown that the CUDA program can expedite the computational process by 13 times when comparing with the traditional sequential calculations.

The bioenergetics fish growth model of Nile Tilapia (Oreochromis niloticus) cultured in the cages in wastewater oxidation pond is presented. The oxidation pond is located in the very successful project area “The Leam Phak Bia Environmental Study Research and Development Project under Royal Initiatives Petchaburi Province” in Thailand. There are five oxidation ponds connected in series in the research area. The measurement of Nile tilapia growth is taken in the third pond in which the water quality is in the effluent standard. The water quality and fish weight are measured in each month for one year period. The fish growth model in the form of ordinary differential equation is introduced to understand the behavior of fish growth due to various environmental factors which are dissolved oxygen demand (DO), water temperature, concentration of plankton, ammonia and biochemical oxygen demand (BOD). In this presented model, we have introduced a new form of food assimilation efficiency as a function of fish weight. It is found that the predicted fish weight obtained from the model is in good agreement with the measurements. Also, this presented model can be applied to predict the fish weight in a wastewater stabilization pond when environmental factors have been changed.

The mathematical model for dissolved oxygen interaction with biochemical oxygen demand in an open channel flow is presented in this paper. We consider unsteady flow in one dimension. The model is in the form of partial differential equations which require to solve the dissolved oxygen coupled with the biochemical oxygen demand at the same corresponding position and time. For some specific flow cases, unsteady flows approach to certain stationary solutions which are possible to derive some corresponding exact solutions. However, for complicated flow problems, analytical solution cannot be derived. Efficient and accurate numerical method is required to solve numerically such the model. In this paper, we present the finite element method with linear basis function for solving the system. The accuracy of numerical solutions have been observed and compared with some existing exact solutions for some steady flow cases. To show the ability of our presented numerical scheme, the interaction between the dissolved oxygen and the biochemical oxygen demand is simulated for unsteady flow cases. Furthermore, the impacts of flow parameters such as flow velocity, kinetic reaction order, and diffusion coefficient have been simulated to observe the dissolved oxygen and biochemical oxygen demand dynamics.

A modified predictor-corrector scheme combining with the depth gradient method (DGM) and the weighted average flux (WAF) method has been presented to solve the one-dimensional shallow water equations with source terms. Approximate solutions in the predictor step are obtained by the DGM with piecewise-linear reconstructions in each cell volume. The source terms can then be calculated directly by these predicted values at the corresponding half-time step. In the corrector step, the TVD version of the WAF method is applied to calculate the numerical fluxes at the same half-time step for each cell face. The accuracy of numerical solutions is shown by applying the method to solve various test cases in both steady and unsteady problems with and without source terms. It shows that the numerical results are in good agreement with the existing analytical solutions as well as experimental data in some test cases.

This paper presents the Lattice Boltzmann method (LBM) for two dimensional shallow water equations to simulate dam break problem. We implemented sequential code and compared it with the exact solution. Then, we implemented three parallel programs with the Compute Unified Device Architecture (CUDA). The first parallel version is a straightforward implementation. The second parallel version reduces the calculations of distribution function in the LBM. The third parallel program has fixed the branch divergence problem with branch distribution. Speed up for the third parallel program is increased approximately 2.1x from the first parallel program, and it is increased approximately 22.7x from the sequential program version.

We consider steady free surface two-dimensional flow due to a localized applied pressure distribution under the effects of both gravity and surface tension in water of constant depth, and in the presence of a uniform stream. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. The behavior of the forced nonlinear waves is characterized by three parameters: the Froude number, F, the Bond number, τ > 1/3, and the magnitude and sign of the pressure forcing parameter ɛ. The fully nonlinear wave problem is solved numerically by using a boundary integral method. For small amplitude waves and F 1/3, there exists both elevation and depression waves. In some cases, a limiting configuration in the form of a trapped bubble occurs in the depression wave solutions.

We consider steady free surface two-dimensional flow due to a localized applied pressure distribution under the effects of both gravity and surface tension in water of constant depth, and in the presence of a uniform stream. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. The behavior of the forced nonlinear waves is characterized by three parameters: the Froude number, F, the Bond number, τ > 1/3, and the magnitude and sign of the pressure forcing parameter ɛ. The fully nonlinear wave problem is solved numerically by using a boundary integral method. For small amplitude waves and F 1/3, there exists both elevation and depression waves. In some cases, a limiting configuration in the form of a trapped bubble occurs in the depression wave solutions.

Incompressible flow simulations governed by the two-dimensional Navier-Stokes model are presented in this work. This set of nonlinear equations is approximated by the lattice Boltzmann method (LBM). Numerical method is developed and implemented in both sequential and CUDA programs. The accuracy of numerical scheme is investigated by simulating two-dimensional duct flows that analytical solution exists. CUDA is used to improve computational performance. Speed up is increased approximately 10x from a sequential program.

The Wavelet-Galerkin finite element method for solving the one-dimensional heat equation is presented in this work. Two types of basis functions which are the Lagrange and multi-level wavelet bases are employed to derive the full form of matrix system. We consider both linear and quadratic bases in the Galerkin method. Time derivative is approximated by polynomial time basis that provides easily extend the order of approximation in time space. Our numerical results show that the rate of convergences for the linear Lagrange and the linear wavelet bases are the same and in order 2 while the rate of convergences for the quadratic Lagrange and the quadratic wavelet bases are approximately in order 4. It also reveals that the wavelet basis provides an easy treatment to improve numerical resolutions that can be done by increasing just its desired levels in the multilevel construction process.

The POD-assisted projective integration method based on the equation-free framework is presented in this paper. The method is essentially based on the slow manifold governing of given system. We have applied two variants which are the “on-line" and “off-line" methods for solving the one-dimensional viscous Bergers- equation. For the on-line method, we have computed the slow manifold by extracting the POD modes and used them on-the-fly along the projective integration process without assuming knowledge of the underlying slow manifold. In contrast, the underlying slow manifold must be computed prior to the projective integration process for the off-line method. The projective step is performed by the forward Euler method. Numerical experiments show that for the case of nonperiodic system, the on-line method is more efficient than the off-line method. Besides, the online approach is more realistic when apply the POD-assisted projective integration method to solve any systems. The critical value of the projective time step which directly limits the efficiency of both methods is also shown.

We have investigated the POD-assisted projective integration method for solving the Navier-Stokes equation based on the equation-free framework. The demonstration of the method is done on the flow past a two-dimensional airfoil(NACA0012) at the Reynolds number of 800. The efficiency of the method in reducing the computational time while maintaining satisfactory accuracy in both transient and periodic solutions is discussed.

This chapter concerns numerical calculations of steady two-dimensional flows due to a pressure distribution applied on the free surface. Gravity is included in the dynamic boundary condition. This problem can serve as a model of an air-cushion vehicle or an inverse method of solution to the classical ship-wave problem in a canal.

We study the stability of the steady waves forced by a moving localised pressure disturbance in water of finite depth. The steady waves take the form of a downstream wavetrain for subcritical flow, but for supercritical flow there is a solution branch for each specified forcing term, which has two localised solitary-like solutions for each Froude number, a small-amplitude and a large-amplitude solution. Our main purpose is to numerically investigate the stability of these steady waves by simulating the unsteady development from appropriate initial conditions. We use a forced Korteweg-de Vries model valid in the transcritical regime for weak forcing, and a fully nonlinear boundary integral simulation. The simulations using the forced Korteweg-de Vries model are in good agreement with the fully nonlinear simulations when the Froude number is near unity for small pressure forcing, as expected. We find that the steady subcritical downstream wavetrain is stable. For supercritical flow, the small-amplitude soli...

In previous work, we investigated two-dimensional steady gravity-capillary waves generated by a localized pressure distribution moving with a constant speed U in water of finite depth h. Localized solitary waves can only exist in subcritical flows where the Froude number F=U/(gh)1/2 1/3, and the magnitude and sign of the pressure distribution, ϵ. In this paper, we examine the two-dimensional stability of these waves using numerical simulations of the fully nonlinear unsteady equations. The results are favorably compared to analogous numerical solutions of the unsteady forced Korteweg–de Vries equation. We find that for ϵ>0, the small-amplitude steady depression wave is stable whereas the larg...

We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow is irrotational; surface tension at the free surface is neglected. Our concern is with the periodic waves generated downstream of the plate edge. These can be characterized by a depth-based Froude number F and the depth d (draft) of the depressed plate. Previous analytical studies have been restricted to small values of d and subcritical flows F

This work presents a new approach to numerically solve the general linear two-point boundary value problems with Dirichlet boundary conditions. Multilevel bases from the anti-derivatives of the Daubechies wavelets are constructed in conjunction with the augmentation method. The accuracy of numerical solutions can be improved by increasing the number of basis levels, but the computational cost also increases drastically. The multilevel augmentation method can be applied to reduce the computational time by splitting the coefficient matrix into smaller submatrices. Then the unknown coefficients in the higher level can be solved separately. The convergent rate of this method is $2^{s}$ , where $1 \leq s \leq p+1$ , when the anti-derivatives of the Daubechies wavelets order p are applied. Some numerical examples are also presented to confirm our theoretical results.