In this manuscript, we present the development of implicit and implicit-explicit ADER and DeC methodologies within the DeC framework using the two-operators formulation, with a focus on their stability analysis both as solvers for ordinary differential equations (ODEs) and within the context of linear partial differential equations (PDEs). To analyze their stability, we reinterpret these methods as Runge-Kutta schemes and uncover significant variations in stability behavior, ranging from A-stable to bounded stability regions, depending on the chosen order, method, and quadrature nodes. This differentiation contrasts with their explicit counterparts. When applied to advection-diffusion and advection-dispersion equations employing finite difference spatial discretization, the von Neumann stability analysis demonstrates stability under CFL-like conditions. Particularly noteworthy is the stability maintenance observed for the advection-diffusion equation, even under spatial-independent constraints. Furthermore, we establish precise boundaries for relevant coefficients and provide suggestions regarding the suitability of specific schemes for different problem.

翻译：本文在DeC框架下，采用双算子形式构建了隐式及隐-显式ADER与DeC方法，并重点分析了这些方法在求解常微分方程（ODE）及线性偏微分方程（PDE）时的稳定性。为评估稳定性，我们将这些方法重新诠释为龙格-库塔格式，发现其稳定性行为存在显著差异——从A稳定到有界稳定区域不等，具体取决于所选阶数、方法及求积节点。这一特性与其显式对应方法形成鲜明对比。当采用有限差分离散空间项，应用于对流-扩散与对流-色散方程时，冯·诺依曼稳定性分析表明，在满足类似CFL条件下方法保持稳定。尤为值得关注的是，即使空间约束独立存在，对流-扩散方程的稳定性仍得以维持。此外，我们确定了相关系数的精确界限，并针对不同问题提出了具体格式的适用性建议。