An Efficient Method to Solve Thermal Wave Equation

Abstract

In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model in one and two dimensions. The proposed method marches in the time direction block by block and there are several time levels in each block. The global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives. Furthermore, the proposed method is validated by comparing the obtained results with the available analytical ones and also compared with the hybrid technique of differential quadrature method and Runge-Kutta fourth order method.

Calculation of Four-Dimensional Unsteady Gas Flow via Different Quadrature Schemes and Runge-Kutta 4th Order Method

“…RK4 is used to overcome time dependent problems. Differential quadrature method is realized as the unknown function at any grid spacing f and its derivatives are approximated as a weighted sum of full the functional values at certain grids in all calculation domain as follows [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]:…”

“…where A x ir and B x ir are the 1st and 2nd weighting coefficients [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. The computed 1st and 2nd derivatives weighting coefficients are various based on choice of shape function.…”

“…The computed 1st and 2nd derivatives weighting coefficients are various based on choice of shape function. Thus, here is how to get it [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]:…”

“…To overcome time dependent problems, DQM can be combined with Runge Kutta, block marching, and finite difference techniques to discretize the time domain [41][42][43][44]. Salah et al [45][46][47] solved nonlinear partial differential equations by classical DQM with Explicit and Implicit Euler method, Runge-Kutta 4th order (RK4), and Block marching to overcome time dependent problems.…”

“…RK4 technique is a numerically scheme mostly applied to solve Initial Value equations, due to its speed and accuracy requiring less computation of higher-order derivative [45][46][47]. Birken [41] employed Runge-Kutta smoothers to get the solution of unsteady viscous flow problems.…”