AI 中文总结
本研究针对二维Kirchhoff型非线性积分微分方程，构造了两种守恒的Crank-Nicolson型三层时间离散格式，证明了能量守恒，建立了非线性格式的先验界与二阶收敛性，并通过不动点迭代求解，数值实验验证了守恒性与收敛性。
AI 中文摘要
我们研究了一个二维空间中的Kirchhoff型非线性积分微分方程，其系数允许依赖于时间，并为相应的初边值问题构造了时间上的守恒离散化。我们考虑了两种对称的Crank-Nicolson型三层格式，一种为局部线性，另一种为真正非线性，并证明了每种格式都保持了具有常数系数的齐次问题的总机械能的离散模拟。对于非线性格式，我们通过直接处理离散能量，在不借助非线性离散Grönwall不等式的情况下，建立了离散解及其离散速度的一致先验界，常数仍然随时间最终呈指数增长，并且我们证明了在时间上对于解及其一阶时间导数的中心差分近似，都具有局部二阶收敛性。每个时间层产生的非线性系统通过不动点迭代求解：给定前两层满足先验界的迭代值，该方程有唯一解，且一旦时间步长足够小，迭代以几何速率收敛于该解。由于先验界是索引局部的，将它们与该单步求解器交替使用，可以在时间依赖系数的情况下，在那些界成立的局部区间上逐步构造轨迹。数值实验在空间离散化不引入误差的设置中进行，展示了离散不变量的守恒，确认了时间上的二阶精度，并验证了不动点迭代的几何收敛性。
英文摘要
We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial--boundary value problem. We consider two symmetric three-layer schemes of Crank--Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Grönwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.
