AI 中文总结
本研究将对角帕德能量定律扩展至次对角帕德逼近，推导了线性半负问题有理时间离散格式的显式离散能量恒等式，经数值实验验证了其阶数与离散耗散恒等式的正确性。
AI 中文摘要
针对线性半负问题，我们推导了由指数函数的次对角帕德逼近生成的有理时间离散格式的显式离散能量恒等式。本研究将[Z. Sun、Y. Wei和K. Wu发表于《SIAM J. Numer. Anal.》2022年第60卷的论文]中的对角帕德能量定律扩展至次对角家族。核心新内容是与离散能量恒等式中半内积项相关的能量系数矩阵的显式乔列斯基型分解。该分解的构造与证明颇具挑战性，因为矩阵元素是帕德系数的交替和，且三角因子具有奇偶依赖的阶乘结构。我们通过将其简化为标量有理恒等式，并借助有限乘积化简与望远镜求和建立这些恒等式，完成了该分解的证明。结合β系数的抵消，该分解得到精确的离散能量定律，其恢复了线性半负问题的经典无条件收缩性。改编自对角帕德能量定律设置的数值实验验证了预测的阶数并确认了离散耗散恒等式。
英文摘要
We derive an explicit discrete energy identity for rational time discretizations generated by the first-subdiagonal Padé approximants of the exponential for solving linear seminegative problems. This work extends the diagonal Padé energy laws in [Z. Sun, Y. Wei, and K. Wu, SIAM J. Numer. Anal., 60 (2022)] to the first-subdiagonal family. The main new ingredient is an explicit Cholesky-type factorization of the energy coefficient matrix associated with the semi-inner-product terms in the discrete energy identity. The construction and proof of this factorization are nontrivial, since the matrix entries are alternating sums of Padé coefficients and the triangular factor has a parity-dependent factorial structure. We prove the factorization by reducing it to scalar rational identities and establishing them through finite product reductions and telescoping summations. Together with a β-coefficient cancellation, the factorization yields an exact discrete energy law that recovers the classical unconditional contractivity for linear seminegative problems. Numerical experiments adapted from the diagonal Padé energy-law setting illustrate the predicted order and verify the discrete dissipation identity.
