We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to the unit sphere without additional projection steps to enforce the unit length constraint on the solution. We construct these algorithms using the exponential map and spherical linear interpolation (SLERP) formula on the unit sphere. Specifically, we introduce a spherical backward Euler method, a projected backward Euler method, and a second-order symplectic spherical Crank-Nicolson method. While all methods require solving a system of nonlinear equations to advance the solution to the next time step, these nonlinear systems can be efficiently solved using Newton's iterations. We will present several numerical examples to demonstrate the effectiveness and convergence of these numerical schemes. These examples will illustrate the advantages of our proposed methods in accurately capturing the dynamics of stiff systems on unit spheres.

翻译：本文针对单位球面上的刚性微分方程求解提出了隐式积分器。该方法将笛卡尔空间中的标准后向欧拉法与Crank-Nicolson法进行扩展，通过融入单位球面固有的几何约束，无需额外投影步骤即可保证解的单位长度约束。我们利用单位球面上的指数映射与球面线性插值公式构建了这些算法。具体而言，我们提出了球面后向欧拉法、投影后向欧拉法以及二阶辛球面Crank-Nicolson法。虽然所有方法均需通过求解非线性方程组来推进时间步长，但这些非线性系统可采用牛顿迭代法高效求解。我们将通过若干数值算例展示这些数值格式的有效性与收敛性，这些算例将证明所提方法在精确捕捉单位球面上刚性系统动力学特性方面的优势。