On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

Thermoelastic Processes by a Continuous Heat Source Line in an Infinite Solid via Moore–Gibson–Thompson Thermoelasticity

Many attempts have been made to investigate the classical heat transfer of Fourier, and a number of improvements have been implemented. In this work, we consider a novel thermoelasticity model based on the Moore–Gibson–Thompson equation in cases where some of these models fail to be positive. This thermomechanical model has been constructed in combination with a hyperbolic partial differential equation for the variation of the displacement field and a parabolic differential equation for the temperature increment. The presented model is applied to investigate the wave propagation in an isotropic and infinite body subjected to a continuous thermal line source. To solve this problem, together with Laplace and Hankel transform methods, the potential function approach has been used. Laplace and Hankel inverse transformations are used to find solutions to different physical fields in the space–time domain. The problem is validated by calculating the numerical calculations of the physical fields for a given material. The numerical and theoretical results of other thermoelastic models have been compared with those described previously.

A New TVD Scheme for Gradient Smoothing Method Using Unstructured Grids

An improved [Formula: see text]-factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity.