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.
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, [
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 [
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
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, are
Firstly, we present the framework of pseudo arc-length moving mesh schemes for the gaseous detonation problem (
The physical domain for computation is
For the one-dimensional space,
Schematic diagram of the quadrilateral element
The values
Definition of boundary surfaces
Surfaces | Vertexes |
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Schematic diagram of the quadrilateral element
The second-order values
Let
In one-dimensional space,
Schematic diagram of the two-dimensional changed areas
In the three-dimensional space, the control element
Schematic diagram of the control element
Three-dimensional changed area
The solution procedure can be illustrated by the following flowchart.
In practice, it is common to use some temporal or spatial smoothing on the weight function
Because discontinuities may initially exist on boundaries or move to boundaries at a later time, redistribution of boundary points should be made in order to improve the quality of the solution near boundary. For convenience, our attention is restricted to the case in which the physical domain
The perfect parallel strategy can reach the conflicting goals of portability and high performance. Here, we adopt the software architecture of MPI, which is a de facto standard for parallel programming on distributed memory systems [
Schematic diagram of the parallel computation.
In the parallel computation, the whole domain is divided into some blocks, and each block is distributed to different processors. The distribution is such that each processor gets allocated with one block. Because the computational complexity is related to the number of mesh points, the whole spatial domain is divided according to mesh points rather than spatial coordinates. In addition, the number of mesh points for each processor computation is the same as far as possible. The common partitions for whole spatial domain are given in Figure
Partitions for spatial domain. (a) Line partition; (b) surface partition; (c) body partition.
In addition, the array
The size of spatial mesh is different for each processor, which leads to the time step for each processor being different. Thus time step is needed to synchronize in the computation.
For data communication, it is necessary to know neighbor processors for each processor. For line partition, there are two neighbors for current processor to communicate, and if the current processor is on the boundary or its neighbor does not exist, the neighbor processor will be signed
The arrays
Processor communication for surface partition; (a) internal communication; (b) boundary communication.
When the current processor is on the boundary, which is shown in Figure
For body partition, there are
Processor communication for body partition; (a) neighbor processors; (b) communication blocks.
For
For
For
Figure
Boundary processor communication; (a) boundary surface processors; (b) boundary edge processors.
In this section, the numerical convergence and the efficiency for our parallel schemes will be shown. In addition, the practical problem will be considered. In our simulation, the parameters are taken as
Firstly, the numerical accuracy is tested for the one-dimensional Euler equations. A periodic boundary condition is used and the initial conditions are set to be
Errors and convergence orders in the density.
Cells | Fixed mesh | Moving mesh | ||
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Error | Order | Error | Order | |
20 | |
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Comparison of various fixed and moving mesh.
Description | CPU time | Time steps |
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(M1) |
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(M2) |
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The simulation results; (a) solution about density; (b) trajectory of mesh.
This is the two-dimensional example. Firstly, the numerical accuracy is tested with the periodic boundary condition and initial conditions
Errors and convergence orders in the density.
Cells | Fixed mesh |
Moving mesh |
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Error | Order | Error | Order | |
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Parallel efficiency with different processors for line partition.
Number of processors | CPU time | Speedup | Efficiency |
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Parallel efficiency with different processors for surface partition.
Number of processors | CPU time | Speedup | Efficiency |
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The computational domain.
The computational results at time 0.15; (a) solution about density; (b) trajectory of mesh.
The computational results at time 0.3; (a) solution about pressure; (b) trajectory of mesh.
The last example is the three-dimensional problem. The numerical exact test is taken firstly. The periodic boundary condition is used and the initial conditions are set to be
Errors and convergence orders in the density.
Cells | Fixed mesh | Moving mesh | ||
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Error | Order | Error | Order | |
30 |
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Parallel efficiency with different processors for body partition.
Number of processors | CPU time | Speedup | Efficiency |
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The computational domain.
The simulation solutions; (a) solution about density at time 0.15; (b) solution about pressure at time 0.3.
The simulation solutions; (a) a block solutions about density at time 0.15; (b) a piece of solutions about density at time 0.3.
In this paper, we have discussed the parallel computation for pseudo arc-length moving mesh schemes. Different from the traditional Euler numerical scheme, the communications between processors are more complex, which include adjacent processor and diagonal processor. The parallel schemes including line, surface, and body partition are all considered in this work and the pseudocodes and schematic diagram are given. Finally, the numerical examples are shown to illustrate that our parallel schemes are effective and the pseudo arc-length moving mesh schemes are convergent and can be used to capture the shock wave.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (Grant nos. 11390363 and 11532012).