非线性声学中时间分数阶Burgers方程的扩散近似
Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics
AI总结：
本文提出用扩散近似方法处理时间分数阶Burgers方程，通过优化策略确定四次项系数，结合Strang分裂法求解超音速部分和扩散部分，验证了分数衰减对波传播的影响。
摘要：
在非线性声学中，将时间分数阶导数引入Burgers方程以建模非线性波的损耗。该项相当于时间卷积积，对数值建模具有很大惩罚作用。本文采用扩散表示法将分数阶导数表示为连续的记忆变量，这些变量满足局部时间的常微分方程。随后，四次项公式产生一个局部偏微分方程组，适合数值积分。确定四次项系数对于确保系统正确性和扩散近似的计算效率至关重要。为此，显示优化与约束是高效策略。使用Strang分裂法依次求解超音速部分和扩散部分，数值实验评估了数值建模的效率，并展示了分数衰减对波传播的影响。
英文摘要：
A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.
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