Implicit Euler Discretization

• Implicit Euler Discretization is a method that evaluates the right-hand side at the future time step, ensuring unconditional stability for stiff and non-smooth problems.
• The scheme requires solving a nonlinear algebraic equation at each step, typically via Newton-type or fixed-point iterations, which preserves key structural properties.
• It is widely applied in the numerical analysis of ODEs, PDEs, and differential inclusions, making it essential for simulations involving stiff dynamics and nonsmooth phenomena.

Implicit Euler Discretization is a time-stepping scheme for ordinary, partial, and stochastic differential equations and inclusions, characterized by evaluating the right-hand side at the future (implicit) time level. This method is widely applied in the numerical analysis of stiff systems, stochastic and deterministic PDEs, differential inclusions, and control algorithms due to its unconditional stability, robustness in nonsmooth or degenerate regimes, and favorable convergence properties even for low-regularity data or in the presence of nonlinearity.

1. Scheme Formulation and Fundamental Properties

The canonical backward (implicit) Euler step for an ODE $\dot{x} = G(x)$ on a time grid $t_k = k h$ is

$x_{k+1} = x_k + h\, G(x_{k+1}),$

where the right-hand side is evaluated at $x_{k+1}$. For general nonlinear $G$, $x_{k+1}$ is determined by solving a nonlinear equation per time step, typically via Newton-type methods or fixed-point iterations (Naumann, 9 Sep 2024).

For inclusions $\dot{x} \in F(t,x)$ and set-valued $F$, the implicit Euler step becomes

$x_{k+1} \in x_k + h\, F(t_{k+1}, x_{k+1}),$

which is a set-inclusion or variational inequality, requiring special solution approaches (Rieger, 2013, Mordukhovich et al., 2014, Li et al., 28 Sep 2024).

Key characteristics of the implicit Euler scheme:

• A-stability: Unconditional stability for linear and many nonlinear stiff problems; the method damps high-frequency modes and is L-stable for pure diffusion (Rieger, 2013, Naumann, 9 Sep 2024, Guo et al., 27 Feb 2025).
• First-order accuracy in time: The local truncation error is $O(h^2)$, yielding global error $O(h)$, provided $G$ is sufficiently smooth (Naumann, 9 Sep 2024).
• Implicitness: Each update involves solving a nonlinear algebraic problem; in semilinear or quasilinear PDE settings, this typically leads to a sequence of nonlinear (or linear for linear problems) elliptic or parabolic solves per time step (Filipov et al., 2018, Benner et al., 2020).
• Strong monotonicity and well-posedness: Under dissipativity (e.g., one-sided Lipschitz or relaxed one-sided Lipschitz conditions), the scheme admits a unique or at least a compact solution set per step and preserves contractivity and monotonicity properties of the continuous problem (Rieger, 2013, Mordukhovich et al., 2014).

2. Application to Differential Equations and Inclusions

ODEs and Nonlinear Dynamics

Implicit Euler is used for stiff ODEs, nonlinear control systems, and optimization algorithms. In the context of Bregman Lagrangian-based accelerated optimization, the implicit Euler discretization yields unconditionally stable, A-stable methods that retain the accelerated continuous decay rates, at the expense of solving nonlinear systems per step (1908.10426). For general nonlinear ODEs, implicit Euler is the method of choice for robustness, especially in the presence of sharp gradients or stiff source terms (Naumann, 9 Sep 2024).

Differential Inclusions

For non-smooth dynamics, the implicit Euler scheme is formulated as a multivalued map, with solution sets that are nonempty, compact, and, under convexity, viable (Rieger, 2013, Mordukhovich et al., 2014, Li et al., 28 Sep 2024). This structure is essential for modeling mechanical systems with impact constraints, sliding mode control, and nonsmooth contact phenomena. Differential-algebraic inclusions are discretized by implicit Euler to ensure robust handling of set-valued feedback and constraint forces, as implemented in modern admittance control strategies for robotics (Li et al., 28 Sep 2024).

Table: Well-Posedness and Solution Properties

Problem Type	Update Rule	Properties
ODE ($\dot{x}=G(x)$)	$x_{k+1}=x_k + h G(x_{k+1})$	Unique $x_{k+1}$
Inclusion ($\dot{x}\in F$)	$x_{k+1} \in x_k+h F(x_{k+1})$	Compact $\mathcal{S}_{k+1}$
SDE/SPDE (linear/multiplic.)	$X_{k+1} = S X_k +$noise	Mean-square convergence

3. Implicit Euler in PDEs and SPDEs

The implicit Euler method is combined with spatial discretizations (finite differences, finite elements, spectral methods) for PDEs and SPDEs:

• Parabolic PDEs: Unconditionally stable (L-stable) for pure diffusion; allows large time steps without violating stability (Guo et al., 27 Feb 2025, Filipov et al., 2018).
• Stochastic PDEs: Fully implicit (backward) Euler methods are the default for SPDEs with stiff operators and noise. For equations driven by additive or multiplicative noise, convergence rates in root mean square are established as $O(h^2 + \Delta t)$ (additive noise, $C^2$ drift), and $O(h^2 + \Delta t^{1/2})$ (multiplicative noise), under regularity assumptions on the data and the noise (Tambue et al., 2017, Mukam et al., 2019).
• Navier–Stokes Equations: The implicit time Euler scheme combined with finite element spatial discretization yields strong $L^2$-convergence, with rates depending on the viscosity, diffusion coefficient, and noise (additive or multiplicative). With Scott–Vogelius elements and additive noise, a polynomial convergence rate is obtained (Bessaih et al., 2020).

In practice, the implicit step yields a large system of linear or nonlinear equations, often tridiagonal (finite difference, 1D), sparse (FEM), or block-structured (multi-physics). Nonlinearities are handled by Newton or fixed-point iterations (Filipov et al., 2018).

Table: Convergence Rates for SPDEs (FEM + Implicit Euler)

Equation Type	Rate (space)	Rate (time)	Noise Type
Semilinear SPDE	$O(h^2)$	$O(\Delta t)$	Additive, $F\in C^2$
Semilinear SPDE	$O(h^2)$	$O(\Delta t^{1/2})$	Multiplicative
Stochastic Navier–Stokes	$O((\Delta t + h^2)^\alpha)$	--	Additive, Scott–Vogelius

(Tambue et al., 2017, Bessaih et al., 2020)

4. Stability, Energy Estimates, and Error Analysis

The implicit Euler method preserves key structural properties:

• Stability: Unconditionally stable for linear dissipative operators; A-stability for monotone ODEs, SPDEs, inclusions (Rieger, 2013, Naumann, 9 Sep 2024).
• Energy dissipation: The method satisfies discrete analogues of energy or entropy inequalities, even in the presence of regularization, degeneracy, or in nonlinear elastodynamics with convex-in-momentum structure (Roubíček, 26 Jul 2024, Roubíček, 19 Apr 2025).
• Error propagation: In composite schemes (e.g., Rothe's method for SPDEs), it is critical that inexact spatial solves do not pollute the overall temporal convergence; by controlling inexactness proportional to the time step, optimal convergence is retained (Cioica et al., 2015).

For optimization algorithms derived via time-discretization of Euler–Lagrange ODEs, implicit Euler discretization delivers global stability for any step-size (subject to local smoothness), outperforming explicit and explicit-implicit schemes that may become unstable or require tuning of parameters (1908.10426).

5. Nonlinear and Non-smooth PDEs: Regularization and Computational Strategies

The method extends to nonlinear, non-smooth, and degenerate problems via regularization and mixed variational techniques:

• Nonlinear diffusion: Time-discrete nonlinear elliptic or TPBVPs solved via Newton-type methods, with Jacobians constructed symbolically or numerically (Filipov et al., 2018).
• Visco-elastic, polyconvex, and finite strain models: Fully implicit time-stepping paired with spatial regularization (multipolar viscosity, $r$-Laplacian, cutoffs) and convexity arguments enables convergence to weak solutions even for thermomechanical models with inelastic or dissipation terms (Roubíček, 26 Jul 2024, Roubíček, 19 Apr 2025).
• Differential inclusions and set-valued control: The method provides robustness to non-smooth constraints and nonsmooth control laws, e.g., in set-valued sliding mode control and discrete-time admittance control for robotics (Li et al., 28 Sep 2024).

6. Algorithmic Implementation, Computational Cost, and Practical Guidelines

• Solution per step: Each time update involves solving (a) a nonlinear algebraic equation (ODEs/differentiators), (b) a system of linear or nonlinear equations (PDEs/SPDEs), or (c) a set-inclusion/variational inequality (nonsmooth dynamics).
• Jacobians and sensitivity: Symbolic approaches to inverting implicit Euler maps and computing the action of inverse Jacobians achieve optimal $O(m\,n^2)$ complexity for $n$-state ODEs, compared to naive $O(m\,n^3)$ approaches (Naumann, 9 Sep 2024).
• Matrix structure and inversion: For constant-coefficient diffusion, backward Euler leads to tridiagonal systems (FDM, 1D); for higher dimensions or nonlinear non-symmetric operators, the matrix may be sparse or block-structured and require iterative or direct solvers (Guo et al., 27 Feb 2025).
• Time step and mesh selection: For parabolic PDEs, there is no CFL stability constraint, but accuracy requires taming $\Delta t$ and the spatial grid $h$ depending on error targets; systematic guidelines trade off memory, computational time, and accuracy (Guo et al., 27 Feb 2025). For SPDEs, time-convergence may be limited by low noise regularity (e.g., boundary-driven), requiring smaller $\Delta t$.

7. Representative Use Cases and Current Research Directions

• Stochastic and deterministic PDE solvers: Widely used for robust and accurate integration of dissipative systems, especially in quantitative uncertainty quantification and control (Tambue et al., 2017, Mukam et al., 2019, Bessaih et al., 2020, Benner et al., 2020).
• Nonsmooth differential inclusions in mechanics and control: Applied to non-smooth, set-valued feedback, impact/contact constraints, and sliding mode/admittance controllers (Mordukhovich et al., 2014, Li et al., 28 Sep 2024).
• Control-theoretic discretization: For sampled-data and flatness-based controller design, implicit Euler retains key controllability and invertibility properties, ensuring that controller synthesis can proceed analogously to the continuous-time setting (Diwold et al., 2022).
• Optimization and machine learning: Time-discretizations of ODEs underpinning optimization flow have led to implicit Euler-based algorithms with provable stability, relevant for the design of accelerated methods (1908.10426).
• Robust exact differentiation: Control and estimation algorithms benefit from proper implicit Euler discretization that eliminates both bias and chattering in discrete differentiators (Seeber, 3 Apr 2024)

Research continues on:

• Adaptive and inexact implicit time-stepping (e.g., Rothe schemes with adaptive spatial solvers) (Cioica et al., 2015).
• Extensions to high-order (multistep, Runge–Kutta) implicit methods with adaptive time-stepping for stiff SPDEs.
• Discrete-time set-valued algorithmics, proximal implementations, and time-stepping for control and nonsmooth contact in robotics (Li et al., 28 Sep 2024).
• Further characterization of convergence rates and computational trade-offs in complex multiphysics and multi-scale systems (Roubíček, 26 Jul 2024, Roubíček, 19 Apr 2025).

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References:

Key technical results, schemes, and properties as detailed above are supported by (Rieger, 2013, Mordukhovich et al., 2014, Cioica et al., 2015, Tambue et al., 2017, Filipov et al., 2018, Mukam et al., 2019, 1908.10426, Bessaih et al., 2020, Benner et al., 2020, Diwold et al., 2022, Seeber, 3 Apr 2024, Roubíček, 26 Jul 2024, Naumann, 9 Sep 2024, Li et al., 28 Sep 2024, Guo et al., 27 Feb 2025, Roubíček, 19 Apr 2025).