Fast and Accurate Numerical Solution of Allen–Cahn Equation

Simulation speed depends on code structures. Hence, it is crucial how to build a fast algorithm. We solve the Allen–Cahn equation by an explicit finite difference method, so it requires grid calculations implemented by many for-loops in the simulation code. In terms of programming, many for-loops make the simulation speed slow. We propose a model architecture containing a pad and a convolution operation on the Allen–Cahn equation for fast computation while maintaining accuracy. Also, the GPU operation is used to boost up the speed more. In this way, the simulation of other differential equations can be improved. In this paper, various numerical simulations are conducted to confirm that the Allen–Cahn equation follows motion by mean curvature and phase separation in two-dimensional and three-dimensional spaces. Finally, we demonstrate that our algorithm is much faster than an unoptimized code and the CPU operation.

An adaptive wavelet collocation method for the optimal heat source problem

Purpose This paper aims to propose a new adaptive numerical method to find more accurate numerical solution for the heat source optimal control problem (OCP). Design/methodology/approach The main aim of this paper is to present an adaptive collocation approach based on the interpolating wavelets to solve an OCP for finding optimal heat source, in a two-dimensional domain. This problem arises when the domain is heated by microwaves or by electromagnetic induction. Findings This paper shows that combination of interpolating wavelet basis and finite difference method makes an accurate structure to design adaptive algorithm for such problems which usually have non-smooth solution. Originality/value The proposed numerical technique is flexible for different OCP governed by a partial differential equation with box constraint over the control or the state function.

The problem of “anomalous” ion transport in high-current vacuum discharges

This paper gives an exhaustive theoretical description of the so-called “anomalous” ion transport phenomenon that exists in vacuum diodes with cathode plasma emission. The phenomenon is represented by the cathode plasma directed movement towards anode. In general terms, the “anomality” means that metal ions are accelerated towards an electrode with a higher electrostatic potential (anode). The theoretical description is given within the framework of a one-dimensional kinetic model, which is based on the Vlasov-Poisson system of equations for plasma and electrostatic field components. Time-dependent accurate numerical solution of this system describes the process evolution leading to “anomalous” ion transport. It was shown that the “anomalous” transport has fast electrodynamic nature.

Numerical solution of two-dimensional nonlinear fractional order reaction-advection-diffusion equation by using collocation method

In this article, two-dimensional nonlinear and multi-term time fractional diffusion equations are solved numerically by collocation method, which is used with the help of Lucas operational matrix. In the proposed method solutions of the problems are expressed in terms of Lucas polynomial as basis function. To determine the unknowns, the residual, initial and boundary conditions are collocated at the chosen points, which produce a system of nonlinear algebraic equations those have been solved numerically. The concerned method provides the highly accurate numerical solution. The accuracy of the approximate solution of the problem can be increased by expanding the terms of the polynomial. The accuracy and efficiency of the concerned method have been authenticated through the error analyses with some existing problems whose solutions are already known.

Numerical Simulation of Overland Flows Using Godunov Scheme Based on Finite Volume Method

A new numerical model for simulating overland flows has been developed using Godunov scheme based on the two-dimensional fully dynamic shallow water equations (SWEs). There are a number of frequently and partially submerged cells due to steep slopes, coarse meshes and small depth when simulating the surface runoff propagation, which are different from the original hydraulic applications such as flooding. In order to provide an accurate numerical solution for overland flows, the model in this work uses the Roe’s approximate Riemann solver for the calculation of fluxes on the triangulated unstructured grid based on the flow sheet regime, and the bottom slope terms are calculated directly by applying the Green’s theorem. To control the global stability of the model, the semi-implicit discretization method is adopted to deal with the highly nonlinear friction terms. The new model provides more comprehensive calculation capabilities, which are proved by several case studies, and the numerical results match well with analytical solutions, experimental data or results computed by other numerical models.