جستجوی مقالات مرتبط با کلیدواژه « numerical accuracy » در نشریات گروه « زمین شناسی »

Several numerical methods are employed to solve the linearized shallow-water equations describing the propagation of Poincaré waves within a one-dimensional finite domain. An analytical solution to the problem, set off by a discontinuous step like elevation, is known and allows to assess the accuracy and robustness of each method and in particular, their ability to capture the traveling discontinuities without generating spurious oscillations.
    The present work examines and applies the central and non-central compact finite difference schemes for the numerical solution of the governing equations of Poincaré waves. Undoubtedly, the central and non-central compact spatial discretization methods have higher numerical accuracy than the central second-order method, and in places where there is an exact solution, the compact methods have shown that these methods are stable under various applied boundary conditions and three-diagonal and five-diagonal forms can be used according to possible limitations. The fourth-order central compact, the third-order and the fifth-order non central compact methods are employed to carry out the spatial differencing of the governing equations and a fourth-order Runge-Kutta method is used for the temporal discretization. The Runge-Kutta time discretization method of the fourth order is a four-step method. In each step, a value for an assumed function is calculated in an intermediate time step, and in the next step, in the same time step, this value is modified.
    In this research, first, the one-dimensional advection equation, which has an analytical solution, is discretized using the above methods, and the performance and numerical accuracy of the methods are measured. Then, the governing equations of Poincaré waves are numerically solved using the mentioned methods and the results are compared for two initial conditions with discontinuous points. The initial condition of the step function is a smooth condition that produces spurious oscillations but the initial condition of the hyperbolic tangent is a sloping condition in the corners, which produces less oscillations. Finally, the numerical solutions of the central and non-central compact methods are compared with each other and the results are analyzed.
    The central and non-central compact methods work well in detecting and identifying the traveling discontinuities. Among the used methods, the non-central compact method of the fifth order has better performance. Moreover, it has a lower error and a higher numerical accuracy. However, with the increase of grid points, the computational cost of this method increases drastically because the fifth-order non-central compact method is a five-point method, and the matrix of their coefficients forms a five-diagonal matrix which has a great impact on the computational time.

By increasing the computing power of computers, the advantage of high-resolution numerical methods for numerical simulation of the governing equations of fluid flow is further emphasized. Recently, increasing the accuracy of numerical methods used for simulation of fluid dynamics problems, particularly the geophysical fluid dynamics problems (e.g., shallow water equations) has been the subject of many research works.
The compact finite difference schemes can provide a simple way to reach the main objectives in the development of numerical algorithms, i.e., having a low cost on the one hand and a highly accurate computational method on the other hand. These methods have also been used for numerical simulation of some geophysical fluid dynamics problems.
However, by splitting the derivative operator of a l compact centra method into one-sided forward and backward operators, a family of compact MacCormack-type schemes can be derived (Hixon and Turkel, 2000). While these classes of compact methods are as accurate as the original compact central methods used to derive the one-sided forward and backward operators, they need less computational work per grid point.
The present work is devoted to the assessment of the accuracy of different methods. The one-dimensional advection equation with the known analytical solution is employed as a prototype model. Also, the truncation error of the traditional second-order MacCormack scheme, the standard fourth-order compact Mac-Cormack scheme, and a fourth-order compact MacCormack scheme with a four-stage Runge–Kutta time marching method are studied. Furthermore, to be able to examine the accuracy, the Lax–Wendroff, the leap-frog and the Beam–Warming methods combined with the second-order and fourth-order compact finite difference methods for spatial differencing are also used. In addition, the convergence rates of different methods are studied. It can be seen that the convergence rates are in agreement with the theoretical order of convergence.
In this work, the traditional second-order MacCormack scheme (MC2), the standard fourth-order compact Mac-Cormack scheme (MC4) developed by Hixon and Turkel (2000) and a fourth-order compact MacCormack scheme with a four-stage Runge–Kutta time marching method (MCRK4) are used for numerical solution of the unsteady and non-linear Rossby adjustment problem (one- and two-dimensional cases). In the one-dimensional case, a single layer shallow water model is used to study the unsteady and nonlinear Rossby adjustment problem. The conservative form of the two-dimensional shallow water equations is used to study the unsteady and nonlinear Rossby adjustment problem in the two-dimensional case. For both cases, the time evolution of a fluid layer initially at rest with a discontinuity in height filed is considered for numerical simulations.

Increasing the accuracy of numerical methods used for simulation of fluid dynamics problems, particularly the geophysical fluid dynamics problems (atmospheric and oceanic), has been the subject of many research works. Recently, due to the increasing computing power of computers, the advantage of high-resolution numerical methods for numerical simulation of the governing equations of fluid flow is further emphasized. The idea of compact finite difference methods goes back to some works conducted in 1920s and 1940s. However, the pioneering works of Kreiss and Oliger (1972), Hirsh (1975) and Lele (1992) made these methdos popular and showd that compact finite difference methdos can be used as a powerful tool for numerical simulation of fluid dynamics problems appearing in different branches of science. These methods have also been used in numerical simulation of geophysical fluid dynamics problems. Due to the promising performance of compact finite difference methods, application of these schemes to numerical simulations of atmospheric and oceanic flows has increased. The compact finite difference schemes have shown that are able to provide a simple way to reach one of the main objectives in the development of numerical algorithms, i.e., having in our disposal a low-cost and highly-accurate computational method. The compact methods have been used extensively for numerical solution of various fluid dynamics problems. These methods have also been applied to numerical solution of some prototype geophysical fluid dynamics problems (e.g., shallow water equations). Most of the compact finite difference methods are symmetric (usually with a 3- or 5-point stencil) and finding each derivative requires a matrix inversion. Recently, a new class of highly-accurate explicit MacCormack type methods has been introduced for computational fluid dynamic. The compact MacCormack type methods were developed by Hixon and Turkel (2000) to split the derivative operator of the central compact method into two one-sided forward and backward operators. This study is devoted to application of the fourth-order compact MacCormack method for numerical solution of the conservative form of two-dimensional non-hydrostatic and fully compressible Navier Stokes equations governing an inviscid and adiabatic atmosphere. Moreover, the second-order MacCormack method is used to compare the performance of the computations. This enables us to measure some aspects of the computational results (such as efficiency and accuracy). Various aspects of the computations such as discretization of the equations for the interior and boundary points, the details of implementation of boundary conditions for different boundary types (e.g., rigid and open boundaries), time step, grid resolution and dissipation are presented. Since, unlike the second-order MacCormack method, the forward operator in the fourth-order compact MacCormack method for approximation of the first derivative at an arbitrary grid point (e.g, j) is not equal to the backward operator at its adjacent point (i.e., j 1), the application of the fourthorder compact MacCormack method for spatial discretization of the source term in vertical momentum equation in non-hydrostatic models needs special treatment. In this work we have used the conventional second-order MacCormack method (MC2), the standard fourth-order compact MacCormack method (MC4) developed by Hixon and Turkel (2000) and a fourth-order compact MacCormack method with a four-stage Runge-Kutta for time advancing (MCRK4) in our numerical simulations. To evaluate the performance of these methods, two test cases including evolution of a warm bubble, and evolution of a cold bubble in a netural atmosphere were simulated. To simulate cold bubble, the test case presented by Straka et al. (1993) and for simulation of warm bubble, the test case of Mendea-Nunez and carrol (1993) are used. Qualitative and quantitative assessment of the results for different test cases showed the superiority of the MCRK4 and MC4 methods over the MC2 method.

Usually, simplified models, such as shallow water model, are used to describe atmospheric and oceanic motions. The shallow water equations are widely applied in various oceanic and atmospheric extents. This model is applied to a fluid layer of constant density in which the horizontal scale of the flow is much greater than the layer depth. However, the dynamics of a two-dimensional shallow water model is less general than three-dimensional general circulation models but is preferred because of its greater mathematical and computational simplicity. Taking intrinsic complexity of fluids, recently, numerical researches have been focused on highly accurate methods. Especially, for large grid spacing numerical simulation, the use of highly accurate methods have become urgent. This trend led to an interest in compact finite difference methods. The compact finite-difference schemes are simple and powerful ways to reach the objectives of high accuracy and low computational cost. Compared with the traditional explicit finite-difference schemes of the same-order, compact schemes have proved to be significantly more accurate along with the benefits of using smaller stencil sizes, which can be essential in treating nonperiodic boundary conditions. Application of some families of the compact schemes to the spatial differencing in some idealized models of the atmosphere and oceans shows that compact finite difference schemes can be considered as a promising method for the numerical simulation of geophysical fluid dynamics problems. In this research work, the sixth-order combined compact (CCD6) finite difference method was applied to the spatial differencing of f-plane shallow-water equations in vorticity, divergence and height forms (on a Randall's Z grid). The second-order centered (E2S), fourth-order compact (C4S) and sixth-order super compact (SCD6) finite difference methods were also used for spatial differencing of the shallow water equations and the results were compared to the ones from a pseudo-spectral (PS) method. A perturbed unstable zonal jet was considered as the initial condition for numerical simulation in which it breaks up into smaller vortices and becomes very complex. The shallow water equations are integrated in time using a three-level semi-implicit formulation. To control the build-up of small-scale activities and thus potential for numerical nonlinear instability, the non-dissipative vorticity equation was made dissipative by adding a hyperdiffusion term. The global distribution of mass between isolevels of the potential vorticity, called mass error, was used to assess numerical accuracy. The CCD6 generated the least mass error among finite difference methods used in this research. By taking the PS method as a reference, the qualitative and quantitative comparison of the results of the CCD6, SCD6, C4S and E2S, indicated the high accuracy of the sixth-order combined compact finite difference method.

In many numerical simulations of fluid dynamics problems, especially those possessing a wide range of length and time scales (e.g., geophysical flows), low-order numerical schemes are insufficient. Compact finite difference schemes, introduced as far back as the 1930s, have been found to be simple ways of reaching the objectives of high accuracy and low computational cost. Compared with the traditional explicit finite difference schemes of the same order, the compact schemes are more accurate with the added benefit of using smaller stencil sizes, which can be essential when treating non-periodic boundary conditions. In recent years, the number of studies devoted to the application of compact schemes to spatial differencing of geophysical fluid dynamics problems has been increasing. This work focuses on the application of a three-point fourth-order compact finite difference scheme for numerical solutions of bottom gravity current over a slope. The governing equations used to perform the numerical simulation are the vorticity-stream function-salinity formulation of the two dimensional viscous incompressible Boussinesq equations. The details of spatial and temporal discretization of the governing equations are presented. For spatial differencing of the equations, the second-order central and a three-point fourth-order compact finite difference schemes are employed. In addition, the second-order two-stage predictor-corrector leapfrog scheme is used to advance the governing equations in time. Derivation of the consistent boundary condition formulation to generate stable numerical solution without degrading the global accuracy of the computations is also presented. To derive the required numerical boundary conditions for salinity and vorticity fields at lateral, top and bottom boundaries of the computational domain, the fourth-order one-sided (forward and backward) compact relations are used. Two values for the salinity and a fixed value for bottom slope angle are used to perform the numerical simulations. Qualitative comparison of the results indicates better performance of the fourth-order compact scheme with respect to the second-order method. Furthermore, the computed value of the rate of the head growth of the gravity current generated by the fourth-order compact scheme is in agreement with existing numerical results, which indicates the accuracy of simulations in a quantitative manner. For the test cases used to perform the simulations in the present work, it was observed that the values of salinity and vorticity generated by the second-order method on bottom boundary were too noisy. While, values of salinity and vorticity generated by the fourth-order compact scheme, especially on the bottom boundary of computational domain, do not show this property and are more accurate than those generated by the second-order method. In addition, the numerical results show that the fourth-order compact scheme can successfully simulate the formation of vortices in the tail section of the gravity current, while the second-order scheme fails.