数值格式的局部精确修正
Locally exact modifications of numerical schemes
摘要：
提出了一类新的求解常微分方程的指数积分器：对已知数值格式的局部精确修正。局部精确性意味着它们在每一点上都保持原系统的线性化。特别地，局部精确积分器保持所有不动点并且是A-稳定的。我们将这种方法应用于流行的格式，包括欧拉格式，隐式中点规则和梯形法则。我们发现局部精确修改的离散梯度计划（对称离散梯度和坐标增量离散梯度）保持其主要的几何性质：能量积分的精确守恒（任意多维哈密顿系统在正则坐标）。二维非谐振子的数值实验表明，局部精确格式在稳定平衡点附近具有很好的精度，远高于新格式的阶数（局部精确修正有时会提高阶数，但在许多情况下保持不变）.
英文摘要：
We present a new class of exponential integrators for ordinary differential equations: locally exact modifications of known numerical schemes. Local exactness means that they preserve the linearization of the original system at every point. In particular, locally exact integrators preserve all fixed points and are A-stable. We apply this approach to popular schemes including Euler schemes, implicit midpoint rule and trapezoidal rule. We found locally exact modifications of discrete gradient schemes (for symmetric discrete gradients and coordinate increment discrete gradients) preserving their main geometric property: exact conservation of the energy integral (for arbitrary multidimensional Hamiltonian systems in canonical coordinates). Numerical experiments for a 2-dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium, much higher than suggested by the order of new schemes (locally exact modification sometimes increases the order but in many cases leaves it unchanged).
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