Initial And Boundary Conditions Research Articles

A novel numerical manner for non-linear coupled variable order reaction-diffusion equation

In this work, an efficient variable order Bernstein collocation technique, which is based on Bernstein polynomials, is applied to a non-linear coupled system of variable order reaction-diffusion equations with given initial and boundary conditions. The operational matrix of Bernstein polynomials is derived for variable order derivatives w.r.t. time and space. The Bernstein operational matrix and collocation technique are applied to the concerned non-linear physical model to achieve a system of non-linear algebraic equations, which are further solved by using Newton method. A few examples are presented to demonstrate the accuracy and stability of the scheme by comparing L2 and L? norm errors between the obtained numerical solutions and existing solutions. The important feature of this article is the graphical exhibitions of the effects of variable order derivatives on the solutions of the considered non-linear coupled reaction-diffusion equation for different particular cases.

Impact of Initial and Boundary Conditions on Coupled Model Simulations for Bay of Bengal

This study investigates the response of a high resolution coupled physical-ecosystem model simulations to initial and boundary conditions (IBCs) from various sources. For this purpose, we used physical parameters from the data sets World Ocean Atlas (WOA09), CSIRO Atlas of Regional Seas (CARS09) and North Indian Ocean Atlas (NIOA). Evaluating model simulated fields with standard validation data suggests that all three experiments could resolve most of the known surface and subsurface features of the Bay of Bengal (BoB) but with considerable differences in salinity and chlorophyll-a (Chl-a) and negligible differences in temperature among model simulations with various IBCs. The modeled Chl-a is well simulated with WOA09 as compared to NIOA and CARS09 data for climatological model simulations, with a correlation of 0.52 for the BoB. Moreover, the vertical distribution of Chl-a is found to be a function of nutrient supply to the base of the euphotic layer and mixed layer. These results significantly highlight the implicit and explicit use of IBCs for better representation of Chl-a concentration in the BoB from a high-resolution coupled model.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘  ’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, N t , E  and rate of convergence, N t , P  . The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

MHD Flow of a Newtonian Fluid in Symmetric Channel with ABC Fractional Model Containing Hybrid Nanoparticles.

The nanofluid is the novelty of nanotechnology to overcome the difficulties of heat transfer in several manufacturing and engineering areas. Fractional calculus has many applications in nearly all fields of science and engineering, which include electrochemistry, dispersion and viscoelasticity. This paper focused on the heat transfer of a hybrid nanofluid in two vertical parallel plates and presented a comparison between fractional operators. In this paper, the fractional viscous fluid model is considered along with physical initial and boundary conditions for the movement occurrences. The analytical solutions have been obtained via the Laplace transform method for the concentration, temperature and velocity fields. After that, we have presented a comparison between Atangana-Baleanu (ABC), Caputo (C) and Caputo-Fabrizio (CF) fractional operators. The comparison of different base fluids (Water, kerosene, Engine Oil) is discussed graphically with respect to temperature and velocity. The results show that due to the high thermal conductivity of water, temperature and velocity are high. While engine oil has maximum viscosity than water and kerosene, thus temperature and velocity are very low. However, due to the improvement in the thermal conductivity with the enrichment of hybrid nanoparticles, the temperature increased, and since the viscosity also increased, the velocity got reduced. Atangana-Baleanu (ABC) fractional operator provided better memory effect of concentration, temperature and velocity fields than Caputo (C) and Caputo-Fabrizio (CF). Temperature and velocity of water with hybridized nanoparticles were high in comparison to kerosene and engine oil.

Prediction of lightning activity using WRF-ELEC model: Impact of initial and boundary conditions

The impact of initial conditions in predicting lightning activity using the Weather Research and Forecasting (WRF) with the electrification (ELEC) extra package, known as WRF-ELEC model, has been investigated. The severity, frequency, and some physical and dynamical properties of lightning occurred during 11 years (2004–2014) are considered. Four thundercloud events with distinctive characteristics over the Tehran area were chosen for detailed study. The selection process was made based on the observational data received from the Iranian Meteorological Organization (IRIMO), ground-based lightning data from the World Wide Lightning Location Network (WWLLN) and satellite-based lightning data from the Lightning Imaging Sensor (LIS). The WRF-ELEC simulations initialized with the European Centre for Medium-Range Weather Forecasts-ERA Interim (ECMWF-ERA Interim), National Centers for Environmental Prediction Final Analysis (NCEP-FNL) and the NCEP operational Global Forecast System (GFS) data have been conducted to obtain the Number Of Lightning flash density (NOL) associated with each case study. The numerical simulations are compared qualitatively and quantitatively with the observations. Several statistical metrics are used in order to determine the performance of the WRF-ELEC model initialized by various IBCs (initial and boundary conditions) for lightning prediction purposes. The results show that in the most of the studied cases, there is a good agreement between the simulated time-averaged horizontal patterns of the Lightning Potential Index (LPI) obtained from the ERA-Interim-based experiments and the locations of lightning occurrence of WWLLN data as well as LIS observations. Furthermore, GFS-based simulations have a better quantitative performance in the NOL prediction than FNL and ERA-Interim-based simulations regarding the values of Standard Deviation (SD), and centered Root Mean Square Error (RMSE). The results of further statistical analysis using different metrics reveal that ERA-Interim initialization has the best performance of lightning activity prediction.

Towards a high-resolution offshore wind Atlas - The Portuguese Case

An accurate offshore wind resource assessment is a key tool for planning marine wind renewable exploitation. To achieve this goal, without resort to an extensive and costly network of anemometric stations or buoys, it becomes necessary to use the so-called atmospheric mesoscale models. This work presents a high spatial resolution (1x1 km) offshore wind resource Atlas for Portugal and the model calibration steps. During the calibration steps, the most adequate: i) atmospheric parameterizations - physics options, ii) initial and boundary conditions (IBC) meteorological datasets, and iii) data assimilation scheme were achieved through sensitivity tests using the common statistical metrics and hourly simulated/observational data. Results show that the most significant improvements are associated with the IBC dataset and the data assimilation scheme used. Thus, the results show that the assimilation procedures coupled with the new ERA-5 reanalysis dataset reduce significantly the errors of the wind speed and direction, especially the normalized mean square error. This reduction, depending on the different calibration setup, can be above 50%. The new Atlas confirms the previous indicators, Portugal presents a high wind power potential, especially for deep offshore regions.

RIPS, a Statistical Method for Characterizing the Limiting Scenario in a BEPU Approach

Combined approaches are now applied in safety analyses of design-basis accidents. They consist of using best-estimate models in the computer codes together with their estimated uncertainties, and require the most unfavorable initial and boundary conditions (IBCs) to be found with respect to the plant operating conditions. This implies determining first the worst-case scenarios, then predicting the figures of merit (FOMs) that must fulfill safety criteria. Such scenarios can be identified by sensitivity studies on IBCs resulting in an input vector of fixed values to realize a deterministic bounding calculation. However, it is a difficult and time-consuming iterative task especially for complex transients with interactions between parameters. Alternatively, the RIPS (Reduction of the Interval of variation of the Parameters of the Scenario) method has been developed in a best-estimate plus uncertainty approach to find the worst IBCs as a set of reduced ranges of variation of the related inputs, rather than by a vector of discrete values. It defines a critical zone for which the FOM is maximized (or minimized). To this end the RIPS method provides quantitative and graphical outcomes enabling identification of the detrimental (or favorable) ranges of variation of the preponderant IBC parameters. This is done through a statistical analysis of a large set of calculations in which all the input parameters and code model uncertainties are randomly sampled. The RIPS method analyzes the higher (or lower) quantiles of the FOM cumulative density function and determines for each input parameter the critical zone within its variation interval, i.e., where it is the most influential. Correlations between parameters are also detected. This paper describes the RIPS method and demonstrates with several examples its ability to adequately identify the critical zone of the IBC configuration space.

Investigating the impact of atmospheric stability on thunderstorm outflow winds and turbulence

Abstract. Downburst events initialized at various hours during the evening transition (ET) period are simulated to determine the effects of ambient stability on the outflow of downburst winds. The simulations are performed using a pseudo-spectral large eddy simulation model at high resolution to capture both the large-scale flow and turbulence characteristics of downburst winds. First, a simulation of the ET is performed to generate realistic initial and boundary conditions for the subsequent downburst simulations. At each hour in the ET, an ensemble of downburst simulations is initialized separately from the ET simulation in which an elevated cooling source within the model domain generates negatively buoyant air to mimic downburst formation. The simulations show that while the stability regime changes, the ensemble mean of the peak wind speed remains fairly constant (between 35 and 38 m s−1) and occurs at the lowest model level for each simulation. However, there is a slight increase in intensity and decrease in the spread of the maximum outflow winds as stability increases as well as an increase in the duration over which these strongest winds persist. This appears to be due to the enhanced maintenance of the ring vortex that results from the low-level temperature inversion, increased ambient shear, and a lack of turbulence within the stable cases. Coherent turbulent kinetic energy and wavelet spectral analysis generally show increased energy in the convective cases and that energy increases across all scales as the downburst passes.

Boundary problem for the singular heat equation

The scheme for solving of a mixed problem with general boundary conditions is proposed for a heat equation \[a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(\lambda(x)\frac{\partial T}{\partial x}\right)\] with coefficient $a(x)$ that is thegeneralized derivative of a function of bounded variation, $\lambda(x)>0$, $\lambda^{-1}(x)$ is a bounded and measurable function. The boundary conditions have the form $$\left\{ \begin{array}{l}p_{11}T(0,\tau)+p_{12}T^{[1]}_x (0,\tau)+ q_{11}T(l,\tau)+q_{12}T^{[1]}_x (l,\tau)= \psi_1(\tau),\\p_{21}T(0,\tau)+p_{22}T^{[1]}_x (0,\tau)+ q_{21}T(l,\tau)+q_{22}T^{[1]}_x (l,\tau)= \psi_2(\tau),\end{array}\right.$$ where by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $\lambda(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by thereduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. This method allows to reduce solving of proposed problem to solving oftwo problems: a quasistationary boundary problem with initialand boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundaryconditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. The first of these problems is solved through the introduction of the quasiderivative. Fourier method andexpansions in eigenfunctions of some boundary value problem forthe second-order quasidifferential equation $(\lambda(x)X'(x))'+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.

Simulation of Monsoon Depressions Using WRF-VAR: Impact of Different Background Error Statistics and Lateral Boundary Conditions

AbstractIn a variational data assimilation system, background error statistics (BES) spread the influence of the observations in space and filter analysis increments through dynamic balance or statistical relationships. In a data-sparse region such as the Bay of Bengal, BES play an important role in defining the location and structure of monsoon depressions (MDs). In this study, the Indian-region-specific BES have been computed for the Weather Research and Forecasting (WRF) three-dimensional variational data assimilation system. A comparative study using single observation tests is carried out using the computed BES and global BES within the WRF system. Both sets of BES are used in the assimilation cycles and forecast runs for simulating the meteorological features associated with the MDs. Numerical experiments have been conducted to assess the relative impact of various BES in the analysis and simulations of the MDs. The results show that use of regional BES in the assimilation cycle has a positive impact on the prediction of the location, propagation, and development of rainbands associated with the MDs. The track errors of MDs are smaller when domain-specific BES are used in the assimilation cycle. Additional experiments have been conducted using data from the Interim European Centre for Medium-Range Weather Forecasts Re-Analysis (ERA-Interim) as initial and boundary conditions (IBCs) in the assimilation cycle. The results indicate that the use of domain-dependent BES and high-resolution ERA-I data as IBCs further improved the initial conditions for the model leading to better forecasts of the MDs.

First Problem of Stokes for Generalized Burgers' Fluids

The velocity field and the adequate shear stress corresponding to the first problem of Stokes for generalized Burgers’ fluids are determined in simple forms by means of integral transforms. The solutions that have been obtained, presented as a sum of steady and transient solutions, satisfy all imposed initial and boundary conditions. They can be easily reduced to the similar solutions for Burgers, Oldroyd-B, Maxwell, and second-grade and Newtonian fluids. Furthermore, as a check of our calculi, for small values of the corresponding material parameters, their diagrams are almost identical to those corresponding to the known solutions for Newtonian and Oldroyd-B fluids. Finally, the influence of the rheological parameters on the fluid motions, as well as a comparison between models, is graphically illustrated. The non-Newtonian effects disappear in time, and the required time to reach steady-state is the lowest for Newtonian fluids.

Ensemble prediction of Mediterranean high-impact events using potential vorticity perturbations. Part I: Comparison against the multiphysics approach

The western Mediterranean is a very cyclogenetic area and many of the cyclones developed over this region are associated with high-impact weather phenomena that affect the society of the coastal countries. Two ensemble prediction systems (EPSs) based on multiphysics and perturbed initial and boundary conditions (IBC) are designed in order to improve the forecast of these heavy rain episodes. The MM5 mesoscale model nested in the ECMWF forecast fields provides the simulations, run at 22.5 km resolution for a two-day period. The multiphysics ensemble combines different model physical parameterization schemes while the other ensemble perturbs the initial state and boundary forcing of the model with the aid of a PV inversion scheme. A PV error climatology derived from the large-scale fields allows to perturb the ECMWF PV fields using the appropriate error range. The verification procedure indicates that even though both EPSs are skillful, the perturbed IBC ensemble is more proficient than the multiphysics EPS for the 19 Mediterranean cyclonic events with heavy rain considered in the study. Therefore the results show a more dominant role of the uncertainties in the initial and boundary conditions than the model error, although both of them contribute significantly to improve the predictability of Western Mediterranean high impact weather situations.

Simulation of Bay of Bengal Tropical Cyclones with WRF Model: Impact of Initial and Boundary Conditions

An attempt is made to delineate the relative performances and credentials of GFS, FNL, and NCMRWF global analyses/forecast products as initial and boundary conditions (IBCs) to the WRF-ARW model in the simulation of four Bay of Bengal tropical cyclones (TCs). The results suggest that FNL could simulate horizontal advection of vorticity maxima at 850 hPa; hence, the tracks are more realistic with least errors as compared to GFS and NCMRWF. The mean landfall errors for 24-, 48-, and 72-hour forecasts are 73, 41, and 72 km, respectively. The TC intensity is well captured by NCMRWF IBCs, as it could predict 850 hPa vorticity maxima. The 24-hour accumulated rainfall is well simulated with FNL, and equitable threat score is more than 0.2 up to 100 mm with minimum bias.

A Note to the Variable Sorptivity Infiltration Equation

A simplification for the variable sorptivity infiltration equation of Poulovassilis et al. (1989) is proposed. The resulting equation has three parameters Sx, c and K0. From these, Sx and c are considered as fitting parameters and K0 as a physical one. The new empirical infiltration equation is tested for precision, parameter time-dependence and applicability for soil surveys. The test was carried out by comparison with reference solutions i.e. infiltration data obtainedexperimentally, analytically or numerically for two different head conditionsat the infiltration surface. A good agreement is observed for all examinedcases. The dependence of the fitting parameters Sx and c on the initialand boundary conditions, as well as the error that arises by taking intoaccount different values of them, are examined. In fine textured soilsparameter c seems to be very small, so that one can easily suppose that the proposed equation reduces to the well-known Philip's infiltration equation (Philip, 1957).

Stability of Difference Schemes for Nonlinear Time-dependent Problems

AbstractIn the present paper, a priori estimates of the stability in the sense of the initial data of the difference schemes approximating quasilinear parabolic equations and nonlinear transfer equation have been obtained. The basic point is connected with the necessity of estimating all derivatives entering into the nonlinear part of the difference equations. These estimates have been proved without any assumptions about the properties of the differential equations and depend only on the behavior of the initial and boundary conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for the finite instant of time t 6 t0 connected with the fact that the solution of the Riccati equation becomes infinite. is already associated with the behavior of the second derivative of the initial function and coincides with the time of the exact solution destruction (heat localization in the peaking regime). A close relation between the stability and convergence of the difference scheme solution is given. Thus, not only a priori estimates for stability have been established, but it is also shown that the obtained conditions permit exact determination of the time of destruction of the solution of the initial boundary value problem for the original nonlinear differential equation in partial derivatives. In the present paper, concrete examples confirming the theoretical conclusions are given.

Nonlinear geostrophic adjustment in the presence of a boundary

The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e. in a half-plane bounded by a rigid wall) is examined in the framework of a rotating shallow water model, using an asymptotic multiple-time-scale theory based on the assumed smallness of the Rossby number ε. The spatial scale is of the order of the Rossby scale. Different initial states are considered: periodic, ‘step’-like, and localized. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time scales f−1 and (εf)−1, respectively. The slow component is not influenced by the fast one, at least for times t [les ] (fε)−1, and remains close to geostrophic balance. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance and Kelvin waves confined near the boundary.The theory provides simple formulae allowing us to construct the initial profile of the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion being described by the equation of a simple wave. The presence of Kelvin waves does not prevent the fast–slow splitting, in spite of the fact that the frequency gap between the Kelvin waves and slow motion is absent. The possibility of such splitting is explained by the special structure of the Kelvin waves in each case considered.The slow motion on time scales t [les ] (εf)−1 is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an algorithm to determine initial slow and fast fields, and the boundary conditions to any order in ε. For the periodic and step-like initial conditions, the slow component behaves in the usual way, conserving mass, energy and enstrophy. In the case of a localized initial disturbance the total mass of the lowest-order slow component is not conserved, and conservation of the total mass is provided by the first-order slow correction and the Kelvin wave.On longer time scales t [les ] (ε2f)−1 the slow motion obeys the so-called modified quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial and boundary conditions for this equation. This modified equation coincides exactly with the ‘improved’ QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001), in the step-like and localized cases. In the periodic case this equation contains an additional term due to the Kelvin-wave self-interaction, this term depending on the initial Kelvin wave profile.

Vapor flux associated with return flow over the Gulf of Mexico: a sensitivity study using adjoint modeling

The moisture flux associated with a return-flow event in the Gulf ofMexico is studied with the aidof a regional forecastmodel and its adjoint.The adjointmodel allows us to determine the derivativesof some aspect of the model output (a scalar, J) with respect to elements of the control vector(initial and boundary conditions). We choose J to be the northward moisture flux through avertical cross section in the northwestern corner of the Gulf at the time when the moisture surgein the US is a maximum. The sensitivities (VJ) are used in conjunction with a set of optimalperturbations to estimate the impact of each of the perturbations on the moisture flux. The perturbationsare chosen to have the structure of the sensitivities (making them optimal as describedin the text), with magnitudes based on the (assumed) uncertainties in the model’s initial andboundary conditions. The linear impact estimate is simply the projection of the perturbation ontothe sensitivity fields. In complementary fashion, we introduce perturbations into the nonlinearforecast model and decompose the actual impact on J into thermodynamic and dynamic components, in order to better understand the nature of the sensitivity. From this numerical experimentwe conclude the following. J is more sensitive to the temperature field than to the other prognosticvariables, roughly by a factor of 3 or 4, based on the optimal perturbations used here. The sensitivityto each of the 3-D prognostic variables peaks in the lower atmosphere near 85 kPa. The sensitivityof J to the sea surface temperature (Ts, a fixed boundary condition) increases with forecast length.More than 30 h before the verification time, the impact of an optimal perturbation to Ts is comparableto the impact of an optimal perturbation to the whole 3-D air temperature field. The zonesof sensitivity to the Ts are intimately tied to the trajectories of low-level air which terminate at thecross section. The impact of an optimal perturbation to the topography along the Sierra MadreOriental mountains is comparable to the impact of optimal perturbations in each of the 3-Dprognostic variables (except T ).

Nonlinear instability of plane liquid sheets

A nonlinear stability analysis has been carried out for plane liquid sheets moving in a gas medium at rest by a perturbation expansion technique with the initial amplitude of the disturbance as the perturbation parameter. The first, second and third order governing equations have been derived along with appropriate initial and boundary conditions which describe the characteristics of the fundamental, and the first and second harmonics. The results indicate that for an initially sinusoidal sinuous surface disturbance, the thinning and subsequent breakup of the liquid sheet is due to nonlinear effects with the generation of higher harmonics as well as feedback into the fundamental. In particular, the first harmonic of the fundamental sinuous mode is varicose, which causes the eventual breakup of the liquid sheet at the half-wavelength interval of the fundamental wave. The breakup time (or length) of the liquid sheet is calculated, and the effect of the various flow parameters is investigated. It is found that the breakup time (or length) is reduced by an increase in the initial amplitude of disturbance, the Weber number and the gas-to-liquid density ratio, and it becomes asymptotically insensitive to the variations of the Weber number and the density ratio when their values become very large. It is also found that the breakup time (or length) is a very weak function of the wavenumber unless it is close to the cut-off wavenumbers.

Transient Drift-Dominated Unipolar Conduction between Concentric Cylinders and Spheres

A new analysis analogous to recent work for parallel-plateelectrodes is presented using a unipolar ion-mobility conduction modelbetween concentric cylindrical or spherical electrodes. Exact solutionsare obtained for the transient behavior of the bulk electric-field andspace-charge density distributions and for the terminal current-voltage(I-V) time dependence. The analysis is generalized to handle any initialand boundary conditions for any terminal constraints or excitations.Using the method of characteristics, the governing first-orderchargeconservation quasi-linear partial differential equation is converted into aset of first-order ordinary differential equations. It is shown that forcurrent source excitations these equations can be easily integrated,while for voltage-source excitations the equations are in the correctform for the Runge-Kutta method of numerical integration. Thecoaxial cylindrical geometry is emphasized, treating the special cases ofthe charging transients to a step voltage or current from rest underspace-charge limited conditions and the discharging transients ofsystems which are in the dc steady state and instantaneously open- orshort-circuited. The solutions are computer drawn in three dimensionsusing a recently available graphics package.

Mixed Problems for Linear Systems of first Order Equations

A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.