Scientific Programming

Volume 2017, Article ID 5896940, 17 pages

https://doi.org/10.1155/2017/5896940

## Parallel Pseudo Arc-Length Moving Mesh Schemes for Multidimensional Detonation

State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Tianbao Ma; nc.ude.tib@labadam

Received 17 January 2017; Accepted 16 May 2017; Published 12 July 2017

Academic Editor: Piotr Luszczek

Copyright © 2017 Jianguo Ning et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We have discussed the multidimensional parallel computation for pseudo arc-length moving mesh schemes, and the schemes can be used to capture the strong discontinuity for multidimensional detonations. Different from the traditional Euler numerical schemes, the problems of parallel schemes for pseudo arc-length moving mesh schemes include diagonal processor communications and mesh point communications, which are illustrated by the schematic diagram and key pseudocodes. Finally, the numerical examples are given to show that the pseudo arc-length moving mesh schemes are second-order convergent and can successfully capture the strong numerical strong discontinuity of the detonation wave. In addition, our parallel methods are proved effectively and the computational time is obviously decreased.

#### 1. Introduction

Detonation is a type of combustion involving a supersonic exothermic front accelerating through a medium that eventually drives a shock front propagating directly in front of it. Detonations occur in both conventional solid and liquid explosives, [1] as well as in reactive gases [2]. Due to the strong discontinuity, one of the most important challenges in detonation simulations is to capture the precursor shock wave. To capture the strong discontinuity in detonation waves, mesh adaptation is an indispensable tool for use in the efficient numerical solution of this type of problem. The moving mesh method is one of the mesh adaptation methods, which relocate mesh point positions while maintaining the total number of mesh points and the mesh connectivity [3–5]. Particularly, Tang et al. proposed a moving mesh method that contained two parts: physical PDE time evolution and mesh redistribution [6–8]. The physical PDE time evolution and mesh redistribution were alternating, and conservative interpolation was used to transfer solutions from old mesh to new mesh. The method is shown to work well generally for hyperbolic conservation. After that, Ning et al. have improved this method and proposed the pseudo arc-length moving mesh schemes, which can deal with the multidimensional chemical reaction detonation problem [9].

In this paper, we will discuss the parallel computation of pseudo arc-length moving mesh schemes for multidimensional detonation. There are some works about the parallel schemes for Euler numerical scheme. Knepley and Karpeev developed a new programming framework, called Sieve, to support parallel numerical partial differential equation (PDE) algorithms operating over distributed meshes [10]. Morris et al. demonstrated how to build a parallel application that encapsulates the Message Passing Interface (MPI) without requiring the user to make direct calls to MPI except for startup and shutdown [11]. Mubarak et al. present us with a parallel algorithm of creating and deleting data copies, referred to as ghost copies, which localize neighborhood data for computation purposes while minimizing interprocess communication [12]. Wang et al. use the parallel scheme to reduce the cost for adaptive mesh refinement WENO scheme of multidimensional detonation [13]. However, there are no discussions about the parallel computation for moving mesh schemes, which will be of concern in this paper. Different from the traditional Euler numerical scheme, the data communications of moving mesh schemes between processors are more complex, which include physical values and mesh points. Besides, the processor communications for pseudo arc-length moving mesh schemes include adjacent processor and diagonal processor. Here, we adopt the software architecture of MPI. The most important advantages of this model are twofold: achievable performance and portability. Performance is a direct result of available optimized MPI libraries and full user control in the program development cycle. Portability arises from the standard API and the existence of MPI libraries on a wide range of machines. In general, an MPI program runs on distributed memory machines. The processor communications for the parallel computation of pseudo arc-length moving mesh schemes are more complex than the traditional Euler scheme, and it includes adjacent processor and diagonal processor. Here, we will consider three kinds of processor partitions to show our parallel schemes. This article is organized as follows. Section 2 introduces the chemical and physical model. Section 3 presents the numerical scheme. Section 4 is devoted to the parallel computation. Section 5 conducts several numerical experiments to demonstrate our schemes. The paper ends with a conclusion and discussion in Section 6.

#### 2. Governing Equations

Instead of using many real elementary reactions, a two-step model was utilized as the testing model. Two-step reaction model considers a complicated chemical reaction to be an induction and an exothermic reaction. For both induction reaction and exothermic reaction, the progress parameters and are unity at first, then decreases to zero, and decreases until an equilibrium state is reached. The rates and are given as follows [14].where is the mass density, the pressure, the temperature, the gas constant, the heat release parameter, and the constants of reaction rates, and and the activation energies.

In deriving fundamental equations, the gas is assumed to be perfect, nonviscous, and non-heat-conducting. In Cartesian coordinates, governing equations for gaseous detonation problem, including the above chemical reaction, arewhere is the multidimensional vector, is the multidimensional matrix function, and is the chemical reaction source term. For the one-dimensional space,For the two-dimensional space,In the case of three-dimensional space,where , , and are the Cartesian components of the fluid velocity in the , , and directions, respectively. Total energy density is defined asHere is the specific heat ratio.

#### 3. Numerical Method

Firstly, we present the framework of pseudo arc-length moving mesh schemes for the gaseous detonation problem (2). Our adaptive scheme is formed by two independents parts: the evolution of the governing equation and the iterative mesh redistribution.

##### 3.1. Time Evolution of Governing Equations

The physical domain for computation is . Given a partition of the mesh domain and a partition of the time interval . The average on the cell isHere, the sign denotes the length for the region in one dimension, the area for the region in two dimensions, and the volumes for the region in three dimensions. Integrating (2) over , we haveApplying the Green-Gauss’s theorem and rewriting, we havewhere is the outward unit normal vector of boundary external surface . The Lax-Friedrichs flux is defined bywhere . It satisfies the conservation and consistencyLet be a partition of boundary external surface and be the outward unit normal vector of surface . Then we have a semidiscrete scheme of (2)where or , , is the internal or external approximate value at .

For the one-dimensional space, , and is the left (right) point of line cell . Thus, (12) becomesBy the initial reconstruction technique [15] to reset , , , on the edge , we can obtain the second-order accurate spatial discretization:where is a nonlinear limiter function which is used to suppress the possible pseudo oscillation. In our computations, we use van Leer’s limiter [15]For the two-dimensional space, , and the diagram for edges , , of the quadrilateral element is shown in Figure 1.