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A highly accurate backward-forward algorithm for multi-dimensional backward heat conduction problems in fictitious time domains

机译：虚拟时域中多维反向热传导问题的高精度反向算法

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Highlights•A highly accurate backward-forward algorithm for multi-dimensional backward heat conduction problems in fictitious time domain.•BFTIM and FFTIM do not require the selection of parameters, such as the viscosity-damping coefficient, fictitious time step, initial guess value and fictitious terminal time.•The proposed schemes are highly accurate, stable, effective, and insensitive to the final data even with large noise level effects.•The numerical implementations of both schemes are simple and have rapid convergence speeds.AbstractThis paper proposes highly accurate one-step backward-forward algorithms for solving multi-dimensional backward heat conduction problems (BHCPs). The BHCP is renowned for being ill-posed because the solutions are generally unstable and highly dependent on the given data. In this paper, the present algorithm combines algebraic equations with a high-order Lie-group scheme to construct one-step algorithms called the backward fictitious integrate method (BFTIM) and the forward fictitious integrate method (FFTIM). First, the original parabolic equation is transformed into a new parabolic equation of an evolution type by introducing a fictitious time variable. Then, the numerical integration of the discretized algebraic equations must satisfy the constraints of the cone structure, Lie-group and Lie algebra at each fictitious time step. Finally, the algorithms with the minimum fictitious time steps along the manifold of the Lie-group scheme approach the true solution with one step when given an initial guess. In addition, this paper provides a strategy to determine the initial guess, which is the reciprocal relationship of the initial condition (IC) and the final condition (FC). More importantly, the IC and FC can be recovered by the BFTIM and FFTIM according to the relation between the IC and FC, even under large noisy measurement data. Five numerical examples of the BHCP are tested and numerical results demonstrate that the present schemes are more effective and stable. In general, the numerical implementations of the BFTIM and FFTIM are simple and have one-step convergence speeds.
机译： 突出显示 • 用于虚拟时域的多维反向导热问题的高精度反向算法。 < / ce：list-item> • BFTIM和FFTIM做不需要选择参数，例如粘度衰减系数，虚拟时间步长，初始猜测值和虚拟终端时间。 • 建议的方案高度准确，稳定，有效且对最终结果不敏感 • 两种方案的数值实现都很简单，并且收敛速度很快。 摘要 本文提出了一种高度精确的单步前进解决多维向后导热问题（BHCP）的算法。 BHCP因不适而闻名，因为解决方案通常不稳定且高度依赖给定数据。在本文中，本算法将代数方程与高阶李群算法相结合，构造了称为后向虚拟积分法（BFTIM）和前向虚拟积分法（FFTIM）的单步算法。首先，通过引入虚拟的时间变量，将原始的抛物线方程式转换为演化类型的新的抛物线方程式。然后，离散虚拟代数方程的数值积分必须在每个虚拟时间步上满足锥结构，李群和李代数的约束。最后，在给出初始猜测时，沿着李群方案的流形具有最小虚拟时间步长的算法以一个步长逼近真实解。另外，本文提供了一种确定初始猜测的策略，即初始条件（IC）和最终条件（FC）的倒数关系。更重要的是，即使在嘈杂的测量数据下，BFTIM和FFTIM也可以根据IC和FC之间的关系来恢复IC和FC。测试了BHCP的五个数值示例，数值结果表明，该方案更加有效和稳定。通常，BFTIM和FFTIM的数值实现很简单，并且具有一步收敛速度。

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