CTCS Schemes for Second Order Wave Equation: Numerical Results and Spectral Analysis

Abstract

IIn this paper, two finite difference methods are used to solve the one-dimensional second order wave equation with constant coefficients subject to specified initial and boundary conditions. Two numerical experiments are considered. The two methods are Central in Time and Central in Space scheme with second order accuracy in both time and space, abbreviated as CTCS (2,2) and Central in Time and Central in Space scheme with second order accuracy in time and fourth order accuracy in space, abbreviated as CTCS (2,4). Properties such as consistency and stability are studied. We also perform spectral analysis of dispersive and dissipative properties of the two methods. Two numerical experiments are considered, and the numerical results are displayed.

Simulation of the Thermal Behavior of a Photovoltaic Solar Panel Using Recent Explicit Numerical Methods

“…This expression is called the constant neighbor (CNe) formula, which gives a method with first order temporal convergence. [23] Now we assume that the neighboring CFD-cell temperatures change linearly [30] in the time, which is a more realistic assumption than the previous one. By this, the system of ordinary differential equations Equation 10 can be uncoupled in the following way:…”

“…This expression is called the constant neighbor (CNe) formula, which gives a method with first order temporal convergence. [ 23 ]…”

“…There are non-traditional numerical methods that have different, often better behavior than the traditional ones. The socalled UPFD (unconditionally positive finite difference) method guarantees the non-negativity of the solutions, [19,20] The NSFD (non-standard finite difference) schemes proposed originally by Mickens [21] can also preserve positivity [22] and can be more accurate than some standard methods, [23,24] Over the past few years, our research group has created a variety of explicit algorithms that are stable for the heat conduction equation for arbitrary time step sizes. This property has been analytically proven for any number of spatial dimensions and arbitrary time step sizes (see, e.g., [25] and its references).…”

Study of the stability of a meshless generalized finite difference scheme applied to the wave equation

“…This bound ensures the production of stable results for the wave equation; however, as can be seen, it is limited to regular discretizations over a rectangle. Another remarkable example of the application of finite differences to obtain the numerical solution of the wave equation can be found in [10], where some CTCS schemes are presented for solving the 1 + 1D case, with great performance.…”