几类分数阶反常扩散方程的数值分析
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摘要
反常扩散现象在自然界中普遍存在,尤其在某些复杂系统的扩散过程中该现象更为常见.为了更好地解释这一现象,不同的学者提出了不同的工具和理论.过去的二十年,许多研究人员发现分数阶微积分可以更为精确地描述这一现象,并且建立了大量的描述复杂系统中反常扩散输运过程的分数阶反常扩散方程.尽管这些分数阶反常扩散方程更精确地解释了所研究的实际问题,但求得这些方程的解析解是比较困难的.因此,寻求此类方程的数值解是十分有意义的.
本文主要研究了描述反常扩散的时间分数阶(径向)扩散方程、分数阶Klein-Kramers方程、分数阶守恒律的数值求解问题,设计了这几类方程的稳定、有效的数值格式,建立了所给格式的误差估计,研究了这些方程的动力学行为.本文的主要工作包括以下三部分：
第一部分考虑了描述次扩散的两类时间分数阶扩散方程的数值求解问题.首先,设计了时间分数阶扩散方程的两种高阶、容易实施、无条件稳定的正交样条配置格式.然后,给出了求解时间分数阶径向扩散方程的两种隐式差分格式,利用数学归纳法和离散极值原理证明了所给数值格式均为无条件稳定的.并用数值结果和数值模拟验证了理论分析的收敛阶和所给格式的有效性.
第二部分讨论了两类分数阶Klein-Kramers方程的数值解.首先,给出了求解时间分数阶Klein-Kramers方程的有限差分格式,给出了数值格式的稳定性和收敛性的严格证明,随后的数值结果验证了理论分析的正确性.进一步,讨论了Levy分数阶Klein-Kramers方程的数值解,建立了Levy分数阶Klein-Kramers方程的隐式和显式的有限差分格式,借助广义的离散极值原理分析了数值格式的稳定性和收敛性,同时给出了几种提高精度的技巧,且数值结果表明所给方法是有效的.
第三部分讨论了分数阶守恒律的数值算法.首先,给出了具有光滑解的周期分数阶非线性守恒律的半离散Fourier谱方法,详细讨论了半离散格式的误差分析,并利用四阶的积分因子-Runge-Kutta方法来求解半离散后的方程组.数值结果表明该方法空间上可以达到谱精度且时间上为四阶收敛的.接着,对于分数阶守恒律的非光滑初值问题,设计了该方程的分数步方法,数值结果表明所给数值算法对于光滑和间断的初值均是有效的.
Anomalous diffusion is one of the most ubiquitous phenomena in nature, especially present in various complex systems. In order to describe anomalous diffusion more ac-curately, different tools are restored by different scholars. Over the past two decades, many researchers found that the fractional calculus is very suitable for describing this phenomenon. Many kinds of anomalous diffusion equations are established. It is difficult to find the analytical solutions of these equations which are closer to the real word. Hence, it is significant to find the numerical solutions of these equations.
The main objective of this thesis is to investigate the numerical solutions of three kinds of anomalous equations, namely, time fractional (radial) diffusion equations, frac-tional Klein-Kramers equations and fractional conservation laws. Stable and efficient numerical schemes are proposed for these equations and the error estimates of our pro-posed numerical schemes are also established. And the numerical simulations are given to reveal the kinetics behaviors of these equations. The main work of this thesis contains the following three parts.
In the first part, we consider the numerical solutions of two kinds of time fractional diffusion equations which are used to describe subdiffusion. We first develop two kinds of high order, easy for programming, unconditionally stable orthogonal spline collocation schemes to solve the time fractional diffusion equation. Then, we establish two kinds of implicit finite difference schemes for the time fractional radial diffusion equation. Applying the method of mathematical induction and discrete maximum principle, we prove that the given schemes are all unconditionally stable. Some numerical results and physical simulations are presented to confirm the rates of convergence and the robustness of the numerical schemes.
In the second part, we focus on the numerical solutions of fractional Klein-Kramers equations. Firstly, we present the finite difference methods for numerically solving the time fractional Klein-Kramers equation and do the detailed stability and error analysis. The numerical examples are provided to confirm the theoretical results. Furthermore, we discuss the numerical solutions of Levy fractional Klein-Kramers dynamics. Explicit and implicit finite difference schemes are established for this equation. By introducing a gener-alized discrete maximum principle, we carefully check the detailed numerical stability and convergence of the numerical schemes. Some other possible techniques for improving the convergent rate or making the schemes efficient in more general cases are also discussed. Numerical results confirm the effectiveness of our numerical schemes.
In the third part, we discuss the numerical algorithms for fractional conservation laws. We present a semi-discrete Fourier spectral method for a periodic fractional conservation law with smooth solutions. The error estimation of the space semi-discrete scheme is rigorously established. And the fourth-order integrating factor-Runge-Kutta method is used to solve the semi-discrete system. The numerical results further confirm the spectral accuracy in space and fourth-order convergence in time. For the non-smooth case, we design fractional step method to deal with it. The numerical examples show that the proposed methods are effective for both smooth and discontinuous initial values'cases.
本文主要研究了描述反常扩散的时间分数阶(径向)扩散方程、分数阶Klein-Kramers方程、分数阶守恒律的数值求解问题,设计了这几类方程的稳定、有效的数值格式,建立了所给格式的误差估计,研究了这些方程的动力学行为.本文的主要工作包括以下三部分：
第一部分考虑了描述次扩散的两类时间分数阶扩散方程的数值求解问题.首先,设计了时间分数阶扩散方程的两种高阶、容易实施、无条件稳定的正交样条配置格式.然后,给出了求解时间分数阶径向扩散方程的两种隐式差分格式,利用数学归纳法和离散极值原理证明了所给数值格式均为无条件稳定的.并用数值结果和数值模拟验证了理论分析的收敛阶和所给格式的有效性.
第二部分讨论了两类分数阶Klein-Kramers方程的数值解.首先,给出了求解时间分数阶Klein-Kramers方程的有限差分格式,给出了数值格式的稳定性和收敛性的严格证明,随后的数值结果验证了理论分析的正确性.进一步,讨论了Levy分数阶Klein-Kramers方程的数值解,建立了Levy分数阶Klein-Kramers方程的隐式和显式的有限差分格式,借助广义的离散极值原理分析了数值格式的稳定性和收敛性,同时给出了几种提高精度的技巧,且数值结果表明所给方法是有效的.
第三部分讨论了分数阶守恒律的数值算法.首先,给出了具有光滑解的周期分数阶非线性守恒律的半离散Fourier谱方法,详细讨论了半离散格式的误差分析,并利用四阶的积分因子-Runge-Kutta方法来求解半离散后的方程组.数值结果表明该方法空间上可以达到谱精度且时间上为四阶收敛的.接着,对于分数阶守恒律的非光滑初值问题,设计了该方程的分数步方法,数值结果表明所给数值算法对于光滑和间断的初值均是有效的.
Anomalous diffusion is one of the most ubiquitous phenomena in nature, especially present in various complex systems. In order to describe anomalous diffusion more ac-curately, different tools are restored by different scholars. Over the past two decades, many researchers found that the fractional calculus is very suitable for describing this phenomenon. Many kinds of anomalous diffusion equations are established. It is difficult to find the analytical solutions of these equations which are closer to the real word. Hence, it is significant to find the numerical solutions of these equations.
The main objective of this thesis is to investigate the numerical solutions of three kinds of anomalous equations, namely, time fractional (radial) diffusion equations, frac-tional Klein-Kramers equations and fractional conservation laws. Stable and efficient numerical schemes are proposed for these equations and the error estimates of our pro-posed numerical schemes are also established. And the numerical simulations are given to reveal the kinetics behaviors of these equations. The main work of this thesis contains the following three parts.
In the first part, we consider the numerical solutions of two kinds of time fractional diffusion equations which are used to describe subdiffusion. We first develop two kinds of high order, easy for programming, unconditionally stable orthogonal spline collocation schemes to solve the time fractional diffusion equation. Then, we establish two kinds of implicit finite difference schemes for the time fractional radial diffusion equation. Applying the method of mathematical induction and discrete maximum principle, we prove that the given schemes are all unconditionally stable. Some numerical results and physical simulations are presented to confirm the rates of convergence and the robustness of the numerical schemes.
In the second part, we focus on the numerical solutions of fractional Klein-Kramers equations. Firstly, we present the finite difference methods for numerically solving the time fractional Klein-Kramers equation and do the detailed stability and error analysis. The numerical examples are provided to confirm the theoretical results. Furthermore, we discuss the numerical solutions of Levy fractional Klein-Kramers dynamics. Explicit and implicit finite difference schemes are established for this equation. By introducing a gener-alized discrete maximum principle, we carefully check the detailed numerical stability and convergence of the numerical schemes. Some other possible techniques for improving the convergent rate or making the schemes efficient in more general cases are also discussed. Numerical results confirm the effectiveness of our numerical schemes.
In the third part, we discuss the numerical algorithms for fractional conservation laws. We present a semi-discrete Fourier spectral method for a periodic fractional conservation law with smooth solutions. The error estimation of the space semi-discrete scheme is rigorously established. And the fourth-order integrating factor-Runge-Kutta method is used to solve the semi-discrete system. The numerical results further confirm the spectral accuracy in space and fourth-order convergence in time. For the non-smooth case, we design fractional step method to deal with it. The numerical examples show that the proposed methods are effective for both smooth and discontinuous initial values'cases.
引文
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