Implicit Euler Finite-Volume Scheme

• Implicit Euler finite-volume schemes combine backward Euler time integration with finite-volume spatial discretization to ensure unconditional stability and conservation.
• The method employs robust nonlinear solvers, such as Newton’s method, to efficiently handle stiff source terms and nonlinearities while preserving physical bounds.
• Widely applied in multiphysics problems, the scheme guarantees convergence to weak solutions and maintains discrete energy and entropy properties.

An implicit Euler finite-volume scheme denotes a class of methods for solving partial differential equations (PDEs) in which time integration is performed via the backward (implicit) Euler method and spatial discretization is accomplished with finite-volume techniques. This approach is applied across a wide range of models, including nonlinear parabolic equations, hyperbolic conservation laws, degenerate elliptic problems, and multiphysics systems. The primary characteristics include unconditional stability with respect to time-step size, conservation of fluxes across control volume interfaces, robust treatment of stiff source terms and nonlinearities, preservation of physically meaningful bounds, and convergence to weak solutions under suitable regularity assumptions.

1. Mathematical Formulation

Let $U(x,t)$ be a vector or scalar field satisfying a PDE of the generic form

$$\partial_t U + \nabla \cdot \mathcal{F}(U) = \mathcal{S}(U,\nabla U),$$

where $\mathcal{F}$ is the flux and $\mathcal{S}$ the source term, on a domain $\Omega\subset\mathbb{R}^d$. The finite-volume spatial discretization partitions $\Omega$ into non-overlapping control volumes $\{V_i\}$, evaluating cell averages $$U_i^n \approx \frac{1}{|V_i|}\int_{V_i} U(x,t^n)\,dx$$.

The backward Euler time discretization advances the solution via

$$U_i^{n+1} = U_i^n - \Delta t\, \mathcal{R}_i(U^{n+1}),$$

where $\mathcal{R}_i$ encodes the cell residual, which comprises face fluxes, sources, and nonlinear terms—typically all evaluated at the new time level $t^{n+1}$ to yield a fully implicit system (Philip, 2014, March et al., 2017, Moortgat et al., 2016).

The scheme leads, at each time step, to a global nonlinear algebraic system, often written compactly as

$\mathcal{H}(U^{n+1}; U^n) = 0,$

which reflects conservation and implicitly treats both diffusion and advection processes.

2. Discretization Procedures

Spatial Discretization:

The finite-volume approach integrates the PDE over each control volume, converting divergence terms into fluxes across adjacent faces using transmissivity coefficients and face-centered differences. For regular grids, central or upwind stencils are applied; for unstructured meshes, two-point flux approximations (TPFA) assure consistency and sparsity (March et al., 2017, Rajasekaran et al., 19 Dec 2025, Sapountzoglou et al., 2024).

Temporal Discretization:

Implicit Euler employs a first-order backward difference for time stepping,

$\frac{U_i^{n+1} - U_i^n}{\Delta t},$

with all spatial fluxes and sources assembled at the implicit level. This yields unconditional stability (see Section 4) (Philip, 2014, Chainais-Hillairet et al., 2015).

Nonlinear Solvers:

The fully discrete, nonlinear system is iteratively solved, typically via Newton's method, with Jacobians incorporating contributions from both explicit and implicit terms. In cases with strong nonlinearities or nonlocal couplings, fixed-point iterations or decoupling strategies (e.g., treating radiative transfers separately) can be employed (Philip, 2014).

3. Structural and Conservation Properties

Conservation:

Discrete mass conservation is enforced via flux balance across the volume interfaces; for degenerate problems, schemes are constructed to respect monotonicity and global constraints, for instance, enforcing

$\sum_i |V_i| U_i^{n+1} = \sum_i |V_i| U_i^n$

at each step (Cancès et al., 2020).

Positivity Preservation:

Recent developments detail positivity-preserving algorithms for implicit schemes, crucial for physical models like compressible Euler and drift-diffusion systems. Modifications include cellwise linear-residual corrections and subsequent conservation-recovery adjustments to maintain nonnegative density and pressure without reducing time-step sizes, yielding a fully conservative and robust update (Huang et al., 2023, Chainais-Hillairet et al., 2015).

Energy and Entropy Dissipation:

Discrete analogs of energy dissipation and entropy production can be constructed. For phase-field and degenerate Cahn–Hilliard models, the schemes guarantee discrete free-energy decay and entropy control, with algebraic proofs ensuring convergence to thermodynamically consistent weak solutions (Cancès et al., 2020).

4. Stability, Accuracy, and Convergence

Unconditional Stability:

The implicit Euler method provides stability irrespective of the time-step magnitudes, as evidenced by spectral analyses whereby all eigenvalues of the update matrix yield contractive behavior. This is particularly advantageous in stiff regimes, multiphysics coupling, or fractured media with extreme mesh aspect ratios (March et al., 2017, Moortgat et al., 2016, Philip, 2014).

Order of Accuracy:

Standard backward Euler finite-volume schemes are first-order accurate in time and, depending on the spatial discretization, up to second order in space (e.g., TPFA or Scharfetter–Gummel). Leading error contributions for a solution $u_h^n$ are typically $O(\Delta t + h^2)$ (March et al., 2017, Chainais-Hillairet et al., 2015, Sapountzoglou et al., 2024).

Convergence:

Rigorous error estimates demonstrate mean-square convergence rates, with the implicit finite-volume discretization shown to converge uniformly: $$\sup_n \mathbb{E}\|u(t_n) - u_h^n\|_{L^2(\Omega)}^2 \leq C(\Delta t + h^2 + h^2/\Delta t).$$ A plausible implication is that choosing $\Delta t \sim h$ yields $O(h)$ mean-square error (Rajasekaran et al., 19 Dec 2025, Sapountzoglou et al., 2024).

In nonlinear and degenerate models, compactness and discrete entropy methods are used to pass limits in the scheme and establish convergence to weak solutions (Cancès et al., 2020, Chainais-Hillairet et al., 2015).

5. Asymptotic-Preserving and Multiphysics Applications

Asymptotic-Preserving Schemes:

Split implicit-explicit ("IMEX") finite-volume schemes—particularly for stiff source terms in Euler equations at low Mach numbers—achieve asymptotic preservation by implicit discretization of acoustically stiff terms (fast pressure) and explicit treatment of convective fluxes. Projection onto equilibrium manifolds after each step maintains positivity and guarantees consistency in the zero-Mach limit. For example, in the scheme of Thomann–Klingenberg–Puppo–Zenk, the elliptic equation for fast pressure is solved implicitly, while the relaxation source terms treat slow pressure and momentum explicitly, presenting robust AP behavior in the incompressible regime (Thomann et al., 2019).

Multiphyics Coupling:

Fully implicit finite-volume schemes are adapted for highly nonlinear, coupled problems including conductive–radiative heat transfer, corrosion (drift-diffusion-Poisson), and multicomponent multiphase flow in fractured porous media. In each instance, the robust time integration is essential for handling interface conditions, nonlocal integral operators, or pressure-velocity coupling. For instance, in the conductive–radiative model, the coupled algebraic system involves nonlinear temperature and radiosity variables, with discrete maximum principles ensuring boundedness and uniqueness (Philip, 2014).

6. Implementation and Computational Aspects

Algorithmic Structure:

• Construct cell and face averages on an admissible mesh (structured or unstructured).
• Assemble backward Euler time-step, evaluate all fluxes and nonlinearities at $t^{n+1}$.
• Apply positivity-preserving corrections (if necessary).
• Solve the resultant algebraic system via Newton, fixed-point, or direct sparse solvers.
• For IMEX or AP schemes, split stiff and non-stiff terms, embedding projection stages as needed.
• Advance to next time-level; repeat until final time.

Efficiency:

Implicit approaches eliminate restrictive CFL-type constraints present in explicit schemes—especially notable for problems with small volumes (as in fracture cells) or stiff physics. Despite a per-step cost increase ($\sim$10–70% overhead in large codes), total computational time is reduced several orders of magnitude compared to explicit methods (Moortgat et al., 2016, Huang et al., 2023).

7. Theoretical Significance and Practical Impact

The implicit Euler finite-volume scheme constitutes a foundational tool for the robust numerical simulation of time-dependent PDEs in fluid dynamics, phase field theory, heat transfer, porous media flow, and related fields. Its properties—including unconditional stability, conservation, positivity, and convergence to weak solutions—enable the treatment of realistic models exhibiting stiff dynamics, sharp interfaces, and nonlocal couplings. Modern variants incorporate second-order extensions, IMEX splitting, positivity-preserving patches, and thermodynamically consistent structures, extending the utility and rigor of this scheme in contemporary computational science (Philip, 2014, Thomann et al., 2019, Cancès et al., 2020, Huang et al., 2023, Rajasekaran et al., 19 Dec 2025, Sapountzoglou et al., 2024, March et al., 2017, Chainais-Hillairet et al., 2015).

References:

(Thomann et al., 2019) Thomann, Klingenberg, Puppo, Zenk (2019): IMEX all-speed relaxation for Euler equations (Philip, 2014) Dohlus, Klingenberg (2014): Fully implicit FV for conductive–radiative heat transfer (Rajasekaran et al., 19 Dec 2025) Rajasekaran, Sapountzoglou (2025): FV convergence for stochastic parabolic PDE (Sapountzoglou et al., 2024) Sapountzoglou, Zimmermann (2024): Error rates for semi-implicit FV in stochastic heat equation (March et al., 2017) March, Carr (2017): FV for multilayer diffusion and unconditional stability (Huang et al., 2023) Huang, Ren, Wang (2024): Positivity-preserving algorithms for implicit FV Euler and NS (Chainais-Hillairet et al., 2015) Chainais-Hillairet, Pierre, Vignal (2015): Implicit Euler + Scharfetter–Gummel FV for corrosion (Cancès et al., 2020) Cancès, Matthes, Nabet (2020): Fully implicit FV for phase-field degenerate Cahn–Hilliard (Moortgat et al., 2016) Moortgat et al. (2016): FV implicit transport methods in fractured porous media