In this paper a new method of numerically solving ordinary differential equations is presented. This method is based on the Gaussian numerical integration of different orders. Using two different orders for numerical integration, an adaptive method is derived. Any other numerical solver for ordinary differential equations can be used alongside this method. For instance, Runge–Kutta and Adams–Bashforth–Moulton methods are used together with this new adaptive method. This method is fast, stable, consistent, and suitable for very high accuracies. The accuracy of this method is always higher than the method used alongside with it. Two applications of this method are presented in the field of satellite geodesy. In the first application, for different time periods and sampling rates (increments of time), it is shown that the orbit determined by the new method is—with respect to the Keplerian motion—at least 6 and 25,000,000 times more accurate than, respectively, Runge–Kutta and Adams–Bashforth–Moulton methods of the same degree and absolute tolerance. In the second application, a real orbit propagation problem is discussed for the GRACE satellites. The orbit is propagated by the new numerical solver, using perturbed satellite motion equations up to degree 280. The results are compared with another independent method, the Unscented Kalman Filter. It is shown that the orbit propagated by the numerical solver is approximately 50 times more accurate than the one propagated by the Unscented Kalman Filter approach.

中文翻译：

---

大地科学中常微分方程的自适应高斯数值积分法数值解

本文提出了一种数值求解常微分方程的新方法。该方法基于不同阶数的高斯数值积分。使用两个不同的阶进行数值积分，得出一种自适应方法。除此方法外，还可以使用任何其他用于常微分方程的数值求解器。例如，Runge–Kutta和Adams–Bashforth–Moulton方法与这种新的自适应方法一起使用。此方法快速，稳定，一致，适用于非常高的精度。此方法的准确性始终高于与其一起使用的方法。该方法在卫星测地学领域有两个应用。在第一个应用中，对于不同的时间段和采样率（时间增量），结果表明，就开普勒运动而言，新方法确定的轨道比具有相同程度和绝对公差的龙格-库塔和亚当斯-巴什福思-莫尔顿方法分别准确度高至少6倍和25,000,000倍。在第二个应用中，讨论了GRACE卫星的实际轨道传播问题。新的数值解算器使用高达280度的受干扰卫星运动方程式传播了该轨道。将结果与另一种独立方法Unscented Kalman滤波器进行了比较。结果表明，数值解算器传播的轨道比无味卡尔曼滤波方法传播的轨道精确约50倍。具有相同程度和绝对公差的Runge–Kutta方法和Adams–Bashforth–Moulton方法。在第二个应用中，讨论了GRACE卫星的实际轨道传播问题。新的数值解算器使用高达280度的受干扰卫星运动方程式传播了该轨道。将结果与另一种独立方法Unscented Kalman滤波器进行了比较。结果表明，数值解算器传播的轨道比无味卡尔曼滤波方法传播的轨道精确约50倍。具有相同程度和绝对公差的Runge–Kutta方法和Adams–Bashforth–Moulton方法。在第二个应用中，讨论了GRACE卫星的实际轨道传播问题。新的数值解算器使用高达280度的受干扰卫星运动方程式传播了该轨道。将结果与另一种独立方法Unscented Kalman滤波器进行了比较。结果表明，数值解算器传播的轨道比无味卡尔曼滤波方法传播的轨道精确约50倍。将结果与另一种独立方法Unscented Kalman滤波器进行比较。结果表明，数值解算器传播的轨道比无味卡尔曼滤波方法传播的轨道精确约50倍。将结果与另一种独立方法Unscented Kalman滤波器进行比较。结果表明，数值解算器传播的轨道比无味卡尔曼滤波方法传播的轨道精确约50倍。