Purpose

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

Design/methodology/approach

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

Findings

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

Originality/value

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

中文翻译：

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微流体液滴集合的一维模型中平滑、尖头和尖锐的冲击波

目的

本文的目的是通过分析和数值方法确定微流体液滴集合、非饱和流中的水流、渗透等一维模型的平滑、尖峰和尖锐冲击波解的存在性，作为幂的函数对流和扩散通量以及上游边界条件；数值研究两种不同初始条件下波的演变；并评估几种有限差分方法的准确性，用于求解控制模型的退化、非线性、对流扩散方程。

设计/方法/方法

常微分方程理论和几种显式的有限差分方法，对对流项使用一阶和二阶，准确的逆风离散，中心离散和紧离散，用于确定稳定传播的波和波的演化的解析解分别从双曲正切和分段线性初始条件到稳定传播的波的前沿。分析确定半离散方案的幅度和相位误差，并评估离散方法的准确性。

发现

对于非零上游边界条件，从分析和数值上都发现，冲击波是平滑的，其陡度随着扩散项功率的增加和上游边界值的减小而增加。对于零上游边界条件，根据对流和扩散项的功率，可能会遇到平滑、尖峰和尖锐的冲击波。对于线性扩散通量，冲击波是平滑的，而对于二次扩散通量，冲击波呈现出尖峰锋，其左空间导数随着对流项功率的增加而减小。对于更高的非线性扩散通量，观察到尖锐的冲击波。随着对流和扩散项的功率增加，波速会降低。双曲正切和分段线性初始条件的解演化表明，波回迅速适应其最终稳定值，而波前需要更长的时间，尤其是对于分段线性初始条件，但稳定波剖面和速度是独立的的初始条件。还表明，非线性扩散通量的离散化在对流项的一阶和二阶逆风离散化的准确性方面比后者的保守或非保守离散化起着更重要的作用。对流项的二阶迎风离散化和紧致离散化显示在波前的底部出现振荡，在那里解为零，但其左空间导数最大。得到的结果是保守的，

原创性/价值

提出了一种新的一维模型，用于微流体液滴传输、不饱和流中的水流、渗透等，包括高阶对流通量和简并扩散，并在分析和数值上进行了研究。其平滑、尖峰和尖锐的冲击波解已被分析确定为非线性对流和扩散通量的功率以及边界条件的函数。这些解用于评估几种有限差分方法的精度，这些方法在空间中使用不同的精度等级以及对流和扩散通量的不同离散化，并可用于评估其他一维、退化的数值程序的精度。 , 对流-扩散方程。