PREM：并行求解非线性演化方程行波解的软件包

摘要: 本文提出了一种新的构造非线性演化方程行波解的并行算法.我们在Maple 18上实现了该算法.通过设计并行算法并使用负载均衡技术，其中的软件PREM的计算效率明显高于已有的串行软件.且基于因式分解算法和运行时间限制，PREM可以自动推导出一些串行程序算不动的复杂方程的部分精确解.相比于已有的其他程序，PREM可自动推导出更多类型的精确行波解.此外，PREM具有灵活的接口和输出.
- 非线性演化方程 /
- 行波解 /
- Riccati方程方法 /
- 并行算法 /
- 负载均衡
Abstract: In this paper, a new parallel algorithm and its implementation called PREM for solving nonlinear evolution equations are presented. PREM is developed in Maple 18. By using parallel and load balancing techniques, PREM is more efficient than any previous serial programs. Furthermore, for some complicated equations that serial programs failed to solve, PREM may obtain some exact traveling wave solutions by factoring algorithm and time limit. In addition, the interface and output of PREM is flexible and diverse. More types of exact travelling wave solutions could be obtained by using this parallel program.
- nonlinear evolution equation /
- travelling wave solutions /
- Riccati equations methods /
- parallel algorithm /
- load balance

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Fig. 1 The parallel structure of the algorithm

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Tab. 1 10 examples to compare the computation efficiency

id	equation
1	$ {u_t} + u{u_x} + p{u_{xxx}} = 0$
2	$q{u^3}-qr{u^2}-q{u^2} + pu{u_x} + qru + {u_t}-{u_{xx}} = 0 $
3	${u_t}-{u_{xx}}-\frac{{u_x^2}}{u} = u\left( {1-{u^3}} \right) $
4	$2qu{u_{xx}} + 2qu_x^2 + p{u_{xx}} + r{u_{xxxx}} + {u_{tt}} = 0 $
5	${u_t} + {u^2}{u_x} + {u_{xxx}} + {u_{yyx}} + {u_{zzx}} = 0 $
6	$q{u^3}-q{u^2} + pu{u_x} + qru + {u_t}-{u_{xx}} = 0 $
7	${u_t} + {v_x} + u{u_x} + q{u_{xxt}} = 0, {v_t} + v{u_x} + u{v_x} + p{u_{xxx}} = 0 $
8	${u_t} + p{v_x} + q{u^2}{u_x} + r{u_{xxx}} = 0, {v_t} + sv{u_x} + u{v_x} + hv{v_x} = 0 $
9	$\begin{array}{l} {u_t}-\frac{{{u_{xx}}}}{2}-\frac{{{v_{xx}}}}{2}-\left( {5 - p} \right)u{u_x} - \left( {1 + p} \right)v{u_x} - pu{v_x} - \left( {2 - p} \right)v{v_x} = 0\\ {v_t} - \frac{{{u_{xx}}}}{2} - \frac{{{v_{xx}}}}{2} - \left( {2 - p} \right)u{u_x} - pv{u_x} - \left( {1 + p} \right)u{v_x} - \left( {5 - p} \right)v{v_x} = 0 \end{array} $
10	${u_t} + {v_x} + u{u_x} + s{u_{xx}} = 0, {v_t} + v{u_x} + u{v_x} + r{v_{xx}} + p{u_{xxx}} = 0 $

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Tab. 2 Comparison of the running time for the above 10 examples

equation id	solutions number (parallel)	solutions number (serial)	time (parallel)	time (serial)	speed-up ratio
1	15	15	2.281	2.452	1.07
2	24	24	4.819	6.454	1.34
3	12	12	8.130	10.656	1.31
4	15	15	5.139	8.067	1.57
5	16	16	3.351	3.927	1.17
6	32	32	14.999	24.856	1.66
7	15	15	6.233	8.779	1.41
8	14	14	18.999	54.212	2.85
9	25	25	9.130	9.973	1.10
10	26	24	11.160	11.354	1.02

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