非线性波动方程的新数值迭代方法

New numerical iterative method for nonlinear wave equations

提出了一种新的求解非线性波动方程的数值迭代法,它是一种半解析的方法.与完全的数值计算方法扰法相比,它能够考虑各阶谐波的相互作用,且能够满足能量守恒定律.用它研究了非线性声波在液体中的传播性质,结果表明,在微扰法适用的声强范围内迭代法也适用,在微扰法不适用的一个较宽的声强范围内迭代法依然适用.

Nonlinear acoustics is an important branch of acoustics and has important applications in some areas,such as high-intensity focused ultrasound,ultrasonic suspension,acoustic cavitation,acoustic harmonic imaging,and parametric emission array.The solving of nonlinear equations in these fields is very important.Regarding the solution of the wave equation of a nonlinear acoustic system,the methods used at this stage generally include complete numerical calculation method,strict analytical method,and perturbation method.1)For the complete numerical calculation method,it covers the finite element method and the finite difference method.The physical meaning of the solution obtained by this kind of method is not clear,and it is difficult to reveal the physical nature of nonlinear event.And in many cases it will lead to the numerical divergence problems,and it is not suitable for all nonlinear problems.2)For the strict analytical method,it can only deal with nonlinear acoustic problems of very few systems,such as the propagation of nonlinear acoustic waves in an ideal fluid.3)For perturbation method,its advantage is that the method is simple and the physical meaning of the solution is clear,but it is only suitable for dealing with nonlinear effects at low sound intensity.And it takes into consideration only the effect of low-order harmonics on higher-order harmonics,with ignoring its reaction,so it does not satisfy the law of conservation of energy.In this paper,we propose a new,semi-analytical numerical iterative method of solving nonlinear wave equations.It is a form of expanding the sound field into a Fourier series in the frequency domain,realizing the separation of time variables from space coordinates.Then,according to the specific requirements for the calculation accuracy,the high frequency harmonics are cut off to solve the equation.Compared with the results from the complete numerical methods(such as finite element method and finite difference method),the solution from this iterative method has a very clear physical meaning.That is,its solution is a combination of harmonics of all orders.Compared with the perturbation method,it can consider the interaction of various harmonics and can satisfy the law of conservation of energy(provided that the system has no dissipation).It is used to study the propagation properties of nonlinear acoustic waves in liquids.The results show that the iterative method is also applicable in the range of sound intensity where the perturbation method is applicable.In a wide range of sound intensity where the perturbation method is unapplicable,the iterative method is still applicable and satisfies the law of conservation of energy(provided that the system has no dissipation).It is unapplicable only if the sound intensity is extremely loud and strong.And when more high-order harmonics are involved,the calculation time by using the numerical method proposed in this paper does not increase sharply.