Adaptive Crank-Nicolson Scheme

• Adaptive Crank-Nicolson schemes are numerical methods that adjust time steps and stabilization parameters for solving time-dependent PDEs and stochastic ODEs with second-order accuracy.
• They incorporate adaptive error estimators and embedded controls to monitor energy dissipation and ensure robust convergence even in stiff and high-dimensional problems.
• These methods are applied in scenarios ranging from multiphysics simulations to Bayesian inference, enhancing computational efficiency and reliability across varied scientific applications.

Adaptive Crank-Nicolson schemes constitute a class of numerical methods that dynamically adjust time-step size and, in several cases, proposal preconditioners, to optimize stability, accuracy, and computational cost for time-dependent PDEs, stochastic ODEs, or Bayesian inference in high-dimensional function spaces. These schemes leverage the second-order accuracy, stability, and conservation properties of the classical Crank-Nicolson method while introducing embedded error estimates, stability monitors, or adaptation strategies to control step size and/or proposal covariance. Modern adaptive CN methodologies are technically sophisticated, incorporating stabilization operators, error-driven time step controllers, block-centered or nonuniform spatiotemporal meshes, and integration with MCMC samplers. This article surveys mathematical formulation, stabilization strategies, error control, applications, and implementation features, referencing foundational developments and extensions from published research.

1. Mathematical Formulation and Stabilization

Adaptive Crank-Nicolson schemes begin from the classical CN formulation for $u_t = \mathcal{L}(u)$, updating

$$u^{n+1} = u^n + \frac{\Delta t}{2}\left[\mathcal{L}(u^n) + \mathcal{L}(u^{n+1})\right],$$

with the extension to nonlinear or gradient-flow systems requiring stabilization. The SLD-CN (linearly stabilized diffusive Crank-Nicolson) scheme for the Cahn–Hilliard equation exemplifies advanced stabilization:

• Discretize $\phi_t = \gamma\,\Delta\mu$, $$\mu = -\varepsilon\,\Delta\phi + \varepsilon^{-1}f(\phi)$$ on a uniform time grid.
• Add two explicit second-order stabilization terms $-A\tau\Delta\,\delta_t\phi^{n+1}$ and $B\,\delta_{tt}\phi^{n+1}$ in the potential update, resulting in a linear symmetric elliptic system with constant coefficients:

$$\begin{aligned} (\phi_h^{n+1}-\phi_h^n)/\tau,\,\psi) &= -\gamma(\nabla\mu_h^{n+1/2},\nabla\psi),\ (\mu_h^{n+1/2},\varphi) &= \varepsilon(\nabla(\tfrac{3}{4}\phi_h^{n+1}+\tfrac{1}{4}\phi_h^{n-1}),\nabla\varphi) + \varepsilon^{-1}(f(\hat{\phi}_h^{n+1/2}), \varphi)\ &\quad - A\tau(\nabla\delta_t\phi_h^{n+1},\nabla\varphi) + B(\delta_{tt}\phi_h^{n+1},\varphi). \end{aligned}$$

Here $A,B$ are stabilization constants tuned to ensure unconditional discrete energy dissipation and control explicit nonlinear terms (Wang et al., 2020). Similar stabilized SAV/CN block-centered finite difference schemes and block-implicit methods have been constructed for Allen–Cahn and fractional diffusion equations (Li et al., 2018, Liao et al., 2020, Ji et al., 2019).

2. Adaptive Time-Stepping Strategies

Adaptive CN methods employ error-driven controllers or solution-based monitors to adjust $\Delta t$ at each step:

• Embedded error estimators, e.g. $$e_{n+1} = 10(\|\phi^{n+1}-\phi^n\|/\epsilon E_C^{n+1})^2$$, are computed, and

$$\tau_{\mathrm{new}} = \rho \left( \frac{\mathrm{tol}}{e_{n+1}} \right)^{1/2} \tau_{\mathrm{old}}$$

applies with safety $\rho$ and user tolerance (Wang et al., 2020, Li et al., 2018).

• In fractional models, temporal adaptivity follows energy dissipation rates (e.g. $$|\mathcal{E}_\alpha[u^n]-\mathcal{E}_\alpha[u^{n-1}]|/\tau_n$$), with step changes concentrated near singular initial times or rapid metastable transitions (Liao et al., 2020, Ji et al., 2019).
• For stochastic PDEs with blowup or singularities, time step is monitored via arc–length or responsive to source term intensity (e.g. $\tau_n \propto [1 + (L'(t_n))^2]^{-1/2}$ or $\tau_n = C/\max_i g(u_i(t_n),\epsilon_i)$), guaranteeing efficient and stable approach to critical events (Padgett et al., 2019).
• In hybrid explicit-implicit or multirank schemes (e.g. AION+TA), cellwise rank assignment and power-of-two subcycling modulate local steps, with robust flux interpolation at interfaces preserving conservation and second-order accuracy (Muscat et al., 2019).

3. Energy Stability, Convergence, and Discrete Principles

Adaptive CN schemes rigorously preserve energy dissipation laws:

• Modified discrete energies $E_C^n$ or $\mathcal{E}_\alpha[u^n]$ are constructed such that

$$E_{C}^{n+1} - E_C^{n} \le -\text{(dissipative terms)} \le 0$$

under algebraic conditions for stabilization constants or variable time-stepping. This extends to unconditional energy stability on nonuniform grids and variable-step adaptive procedures (Wang et al., 2020, Liao et al., 2020, Ji et al., 2019).

• Maximum principle and monotonicity in nonlinear degenerate systems are preserved via M-matrix structure and positivity/monotonicity-preserving mesh constraints (Padgett et al., 2019, Liao et al., 2020).
• Error analysis affords second-order accuracy in time, spectral or high-order spatial accuracy (via spectral Galerkin, finite element, or block-centered methods), and polynomial control of error prefactors even for stiff regimes or small interface thickness $\varepsilon$ (Wang et al., 2020, Li et al., 2018, Ji et al., 2019).
• Over arbitrary (even stochastically perturbed) spatial and temporal grids, convergence is maintained up to singularities (e.g. quenching time), and adaptive methods allow energy-optimal resolution (Padgett et al., 2019).

4. Adaptive CN for Bayesian and High-Dimensional Inference

For infinite-dimensional Bayesian inference, adaptive Crank-Nicolson preconditioned MCMC algorithms (pCN) and their generalizations present dimension-independent mixing and improved sampling efficiency:

• Standard pCN proposes $v = \sqrt{1-\beta^2}u + \beta w$, $w \sim \mathcal{N}(0, \mathcal{C}_0)$, ensuring reversibility w.r.t.\ Gaussian priors under refinement (Hu et al., 2015).
• Adaptive pCN updates the proposal covariance operator mode-by-mode via sample history, using recursive statistics for leading eigenmodes; this satisfies ergodicity under diminishing adaptation and containment conditions, and delivers rapid drop in autocorrelation, with higher effective sample size for PDE inverse problems (Hu et al., 2015).
• The $t$-preconditioned Crank-Nicolson (tpCN) approach fits a multivariate $t$-distribution to the ensemble at each annealed tempering, with the proposal

$$x' = \mu_s + \sqrt{1-\rho^2}(x-\mu_s) + \rho \sqrt{Z_m} W_m$$

($W_m \sim \mathcal{N}(0, C_s)$, $Z_m$ from Gamma) yielding improved mixing for non-Gaussian posteriors. Sequential Kalman Tuning (SKT) combines EKI initialization and tpCN correction for each temperature schedule, outperforming standard SMC or pCN for gradient-free inference in high dimensions (Grumitt et al., 10 Jul 2024).

5. Spatial Adaptivity, Conservation, and Multigrid Implementation

Adaptive Crank-Nicolson can be extended to frameworks combining spatial and temporal adaptivity:

• In AMR contexts, implicit CN solvers are used for parabolic terms on block-structured grids, with synchronous time stepping or local stepping depending on physical stability limits, supported by robust refluxing and prolongation at coarse-fine interfaces to maintain second-order spatial accuracy (Matsumoto, 2010).
• Multigrid solvers (FMG, V-cycle, FAS) handle the inversion of implicit elliptic systems, ensuring rapid convergence even for very small diffusive time steps on deep AMR hierarchies (Matsumoto, 2010).
• For hybrid explicit–implicit integration (e.g. LES-RANS coupling), cellwise flux assembly and conservative interface reconstruction secure total mass, momentum, and energy conservation across strongly heterogeneous mesh ranks (Muscat et al., 2019).

6. Numerical Experimentation and Practical Performance

Adaptive CN schemes are validated through a range of canonical and complex problems:

• Cahn–Hilliard, Allen–Cahn, and molecular beam epitaxy problems demonstrate the capacity of adaptive SLD-CN and SAV/CN methods to handle multiscale coarsening, singularities, and metastable dynamics, preserving energy curves and interface morphology even with orders-of-magnitude reduction in time steps compared to uniform methods (Wang et al., 2020, Li et al., 2018, Ji et al., 2019, Liao et al., 2020).
• MHD Ohmic dissipation, wave packet advection, shock tube, isentropic vortex transport, and stochastic quenching equations established robust convergence, monotonicity, and efficiency against reference solutions for a variety of diffusive and hyperbolic PDEs under adaptive or hybrid discretizations (Matsumoto, 2010, Muscat et al., 2019, Padgett et al., 2019).
• Bayesian inverse problems in high dimension, with KL expansions and PDE-based forward models, showed low bias and fast convergence when using adaptive SKMC-tpCN schemes, especially with normalizing flow preconditioning for strongly non-Gaussian targets (Grumitt et al., 10 Jul 2024).

7. Implementation Aspects and Recommendations

Key features for practical implementation of adaptive Crank-Nicolson schemes include:

• Choice of spatial discretization (Legendre–Galerkin, FEM, finite difference, block-centered, spectral FFT).
• Embedded error control and step size adaptation via local $L^2$ differences, energy dissipation rates, or predictor-corrector strategies.
• Robust assembly of linear (or elliptic) systems, favoring constant/symmetric coefficient matrices for efficient direct or fast transform solvers (O($N\log N$)).
• Avoidance of nonlinear iterations by explicit treatment of nonlinear terms, stabilization via extrapolation and/or auxiliary variables.
• Step-size and covariance adaptation in MCMC using sample statistics, adaptive truncation, and PI-free controllers.
• Practical tuning: safety factors ($0.9$), tolerance ($10^{-3}$), adaptivity bounds, mode truncation for high-dimensional inference, and conservative flux interpolation at mesh interfaces.

Adaptive Crank-Nicolson methods thus form an integral component of contemporary numerical analysis for multiscale, stiff, and high-dimensional dynamical systems, offering rigorous stability, efficiency, and spatial-temporal adaptivity in simulation and inference applications (Wang et al., 2020, Matsumoto, 2010, Hu et al., 2015, Muscat et al., 2019, Li et al., 2018, Liao et al., 2020, Ji et al., 2019, Padgett et al., 2019, Grumitt et al., 10 Jul 2024).