量子冻结-奥辛同伦分析方法与LCHS求解非线性偏微分方程
Quantum Frozen--Oseen homotopy analysis method with LCHS for solving nonlinear partial differential equations
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中文总结 AI 辅助
提出冻结-奥辛量子同伦方法（FOQHAM），通过冻结Fréchet导数并利用LCHS传播，将非线性PDE近似转化为量子线性演化，在多个经典方程上验证了非迭代近似的准确性。
中文摘要 AI 辅助
非线性偏微分方程（PDE）是计算流体动力学的基础，然而在细网格上求解非线性输运问题在计算上仍然具有挑战性。量子线性演化算法为大规模模拟提供了一条可能的途径，但无法直接传播非线性耦合。在此，我们开发了一种冻结-奥辛量子同伦方法（FOQHAM）框架，将考虑输运的辅助算子选择与所得线性表示的规模联系起来。我们在指定的流动剖面上冻结完整的Fréchet导数，保留输运和剖面梯度耦合，并通过乘积提升将每个指定的有限阶同伦层级封闭为固定的仿射线性系统。我们通过哈密顿模拟的线性组合（LCHS）公式化其传播，无需外部同伦迭代或剖面更新。对于空间半离散方程，我们建立了局部近似界和有限阶层级的精确封闭性。在Burgers、Korteweg-de Vries、Zakharov-Kuznetsov以及一维和二维等温可压缩Navier-Stokes方程上的经典测试提供了在所测试范围内精确非迭代近似的数值证据。该框架因此将非线性PDE近似与量子线性演化联系起来，并为量子加速提供了一条有条件路径。
英文摘要
Nonlinear partial differential equations (PDEs) underpin computational fluid dynamics, yet resolving nonlinear transport on fine grids remains computationally demanding. Quantum linear-evolution algorithms offer a possible route to large-scale simulation but cannot directly propagate nonlinear coupling. Here we develop a Frozen--Oseen quantum homotopy method (FOQHAM) framework that links transport-aware auxiliary-operator selection to the size of the resulting linear representation. We freeze the full Fréchet derivative at a prescribed flow profile, retaining transport and profile-gradient coupling and close each prescribed finite-order homotopy hierarchy as a fixed affine linear system by product lifting. We formulate its propagation through Linear Combinations of Hamiltonian Simulations (LCHS), without outer homotopy iterations or profile updates. For spatially semidiscrete equations, we establish local approximation bounds and exact closure of the finite-order hierarchy. Classical tests on Burgers, Korteweg--de Vries, Zakharov--Kuznetsov, and one- and two-dimensional isothermal compressible Navier--Stokes equations provide numerical evidence of accurate non-iterative approximations in the tested regimes. The framework thus connects nonlinear PDE approximation to quantum linear evolution and offers a conditional path toward quantum acceleration.
发表机构
- Beihang University（北京航空航天大学）
- Hangzhou International Innovation Institute of Beihang University（北京航空航天大学杭州国际创新研究院）
机构由 AI 辅助整理，请以论文原文为准。
