Application Of Finite Difference Schemes Research Articles

CFD Julia: A Learning Module Structuring an Introductory Course on Computational Fluid Dynamics

CFD Julia is a programming module developed for senior undergraduate or graduate-level coursework which teaches the foundations of computational fluid dynamics (CFD). The module comprises several programs written in general-purpose programming language Julia designed for high-performance numerical analysis and computational science. The paper explains various concepts related to spatial and temporal discretization, explicit and implicit numerical schemes, multi-step numerical schemes, higher-order shock-capturing numerical methods, and iterative solvers in CFD. These concepts are illustrated using the linear convection equation, the inviscid Burgers equation, and the two-dimensional Poisson equation. The paper covers finite difference implementation for equations in both conservative and non-conservative form. The paper also includes the development of one-dimensional solver for Euler equations and demonstrate it for the Sod shock tube problem. We show the application of finite difference schemes for developing two-dimensional incompressible Navier-Stokes solvers with different boundary conditions applied to the lid-driven cavity and vortex-merger problems. At the end of this paper, we develop hybrid Arakawa-spectral solver and pseudo-spectral solver for two-dimensional incompressible Navier-Stokes equations. Additionally, we compare the computational performance of these minimalist fashion Navier-Stokes solvers written in Julia and Python.

Application of Finite Difference Schemes to 1D St. Venant for Simulating Weir Overflow

Depth averaged equations are commonly used for modelling hydraulics problems. Nevertheless, the model may not be able to accurately assess the flow in the case of different flow regimes, such as hydraulic jump. The model requires appropriate numerical method or other numerical treatments in order to simulate the case accurately. A finite volume scheme with shock capturing may provide a good result, but it is time consuming as compared to the commonly used finite difference schemes. In this study, 1D St. Venant equation is solved using Artificial Viscosity Lax-Wendroff and Mac-Cormack with TVD filter schemes to simulate an experiment case of weir overflow. The case is chosen to test each scheme ability in simulating flow under different flow regimes. The simulation results are benchmarked to the observed experimental data from previous study. Additionally, to observe the scheme efficiency, the simulation time between the models are compared. Therefore, the most accurate and efficient scheme can be determined.

New model for overhead lossy multiconductor transmission lines

A new model for time-domain electromagnetic transient analysis of overhead multiconductor transmission lines with frequency-dependent electrical parameters is presented. The model is based on the method of characteristics, which has been used before by means of the application of finite difference schemes. Conversely to the regular method of characteristics, the model presented here does not require the spatial discretisation along the line. Also, the frequency dependence of the electrical parameters is included by means of a transient resistance matrix. To validate the model, the results are compared to those from a frequency-domain method, the alternative transients program/electromagnetic transients program (ATP/EMTP) and the electromagnetic transients program-restructured version (EMTP-RV).

The efficient prediction of shallow water flows—Part II: Application

Numerical results are presented and compared for the application of finite difference schemes for the shallow water equations on a range of problems.

FINITE DIFFERENCE TREATMENT OF IRREGULAR INTERFACES: AN ERROR ANALYSIS

The application of finite difference schemes to handle a horizontal interface between two media gives numerically adequate results, as evidenced in the solution of parabolic wave equations with density variations. Using the same finite difference schemes for handling a stair-step approximation to an irregular interface gives satisfactory results provided that the slope angle is small, but a small error is introduced. In the event that the slope angle is not small, this error has to be handled carefully since it may influence the results. Using an analysis of the error, this paper derives a closed form expression for a correction term. An irregular interface can then be handled adequately by the same numerical treatment used for the horizontal interface by applying this correction term. A test case is presented to demonstrate the inaccuracy of using one standard numerical horizontal interface treatment and to show how our new development improves the accuracy.