无振荡方法用于具有噪声初始条件的半线性扩散方程
Oscillation-free method for semilinear diffusion equations under noisy initial conditions
AI总结：
本文提出一种无振荡方法，用于解决具有噪声初始条件的半线性扩散方程的初值边界值问题，通过对称Strang分裂分离线性扩散和非线性余项，开发无振荡方案以克服数值求解线性扩散部分时的振荡行为，并通过Fisher型方程验证方法的准确性与实用性。
摘要：
初始条件中的噪声从测量误差产生，可以在数值解中引起不想要的振荡，这些振荡会传播。本文提出一种技术，用于在求解半线性扩散方程的初值边界值问题时禁止此类振荡误差。对称Strang分裂被应用于方程中，以分别求解线性扩散和非线性余项。开发了一种无振荡方案以克服任何振荡行为。为了展示稳定振荡的危害，我们比较了使用加权隐式欧拉方案的方法与Crank-Nicolson方法。通过局部线性化分析无振荡特征和稳定性。通过求解Fisher型方程证明了无振荡方法的准确性，并进一步通过在初始条件存在随机误差时成功产生无振荡解来验证其实用性。
英文摘要：
Noise in initial conditions from measurement errors can create unwanted oscillations which propagate in numerical solutions. We present a technique of prohibiting such oscillation errors when solving initial-boundary-value problems of semilinear diffusion equations. Symmetric Strang splitting is applied to the equation for solving the linear diffusion and nonlinear remainder separately. An oscillation-free scheme is developed for overcoming any oscillatory behavior when numerically solving the linear diffusion portion. To demonstrate the ills of stable oscillations, we compare our method using a weighted implicit Euler scheme to the Crank-Nicolson method. The oscillation-free feature and stability of our method are analyzed through a local linearization. The accuracy of our oscillation-free method is proved and its usefulness is further verified through solving a Fisher-type equation where oscillation-free solutions are successfully produced in spite of random errors in the initial conditions.
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