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Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
Yue Wu, Chi-Wang Shu
TL;DR
Nonlinear degenerate diffusion creates a tension between explicit time-step restrictions and the nonlinear solves required by fully implicit methods. The paper analyzes EIN schemes using projected Kirchhoff transforms and proves energy stability under coefficient bounds and IMEX conditions, without a time-step restriction. The estimates allow degenerate diffusion and extend to fixed-metric systems and consistent positive-weight quadrature.
Problem
Explicit discretization of nonlinear diffusion has a parabolic time-step restriction, while fully implicit treatment generally requires costly nonlinear iterations.
Source: §1.1 · p. 2Method
The analysis applies IMEX-based EIN time stepping to the projected Kirchhoff-transform formulation, with linear diffusion implicit and the nonlinear remainder evaluated at known states.
Source: §1.2 · p. 2, §3.1 · p. 6Results
Under the stated coefficient bounds and IMEX conditions, first- to third-order EIN estimates are unconditional in the time step and require no positive lower bound on the nonnegative diffusion coefficient.
Source: §3.2 · pp. 7–8, §7 · p. 18Takeaways & Limitations
The estimates dissipate the discrete convex energy across a wide range of spatial discretizations and extend to fixed-metric systems and consistent positive-weight quadrature.
Source: §5.1.2 · p. 15, §7 · p. 18The EIN3-ARS(4,4,3) coefficient matrices fail the stated stability criterion, and increasing the stabilization parameter cannot make them satisfy it.
Source: §4.6 · p. 12
Abstract
from arXiv · showhide
We study nonlinear degenerate parabolic equations with state-dependent coefficients admitting a scalar energy potential. The explicit--implicit--null (EIN) method avoids severe time step restrictions of explicit time stepping and costly Newton--Raphson iterations required by a fully implicit treatment of the nonlinearity. Using orthogonal projection of the Kirchhoff transform, we prove nonlinear energy stability for first- to third-order EIN time stepping coupled with a wide range of spatial discretizations that admit self-adjoint positive semidefinite discrete operators. Under an upper bound for the diffusion coefficient on relevant stage segments, two IMEX coefficient conditions yield sufficient stabilization bounds without any time step restriction. The estimates dissipate the original convex energy and allow degenerate diffusion. We also extend the analysis to diffusion systems with a scalar energy potential in a fixed positive definite metric, and to consistent positive-weight quadrature discretizations. Applications include porous-medium flow, nonlinear heat transfer in plasma, and multicomponent diffusion.
1. Introduction.
The paper develops energy-stable EIN time discretizations for nonlinear degenerate diffusion, addressing explicit time-step restrictions and the nonlinear solves of fully implicit methods. Its analysis covers first- to third-order schemes and broad spatial discretizations under coefficient and operator conditions.
1.1. The model problem and energy stability.
- The model permits a nonnegative state-dependent diffusion coefficient that may vanish, and its Kirchhoff formulation yields a convex energy dissipated by smooth solutions.The paper also defines discrete counterparts of one-step and space-time energy stability.Source: §1.1 · pp. 1–2
- Explicit time stepping faces a mesh-dependent parabolic restriction, while fully implicit treatment generally requires costly nonlinear iterations, motivating an approach that avoids both burdens.The paper identifies porous-medium flow, unsaturated flow, and temperature-dependent heat transfer as settings with nonlinear diffusion.Source: §1.1 · p. 2
1.2. The EIN time discretization.
- EIN applies an IMEX split that treats linear diffusion implicitly and evaluates the nonlinear remainder at known states, so implicit stages require linear rather than Newton–Raphson or Picard solves.The paper situates EIN within prior work on adding and subtracting linear diffusion and on IMEX methods.Source: §1.2 · pp. 2–3
1.3. Our contribution.
- First- to third-order EIN schemes dissipate the discrete convex energy without a time-step restriction when the stated coefficient bounds hold, and the analysis extends to fixed-metric systems and consistent positive-weight quadrature.The diffusion coefficient need not have a positive lower bound; the spatial operator must be self-adjoint and positive semidefinite, with orthogonal projection of the Kirchhoff transform.Source: §1.3 · pp. 3–4
2. Spatial discretization.
The spatial analysis relies on a finite-dimensional approximation space and a self-adjoint positive semidefinite discrete Laplacian. Projecting the Kirchhoff transform supports the discrete energy identity and its conditional energy-dissipation estimate.
2.1. Discrete Laplacian.
- The spatial setting requires a self-adjoint positive semidefinite discrete Laplacian, allowing multiple symmetric finite-element, DG, and finite-volume realizations.The analysis does not require a particular basis, element shape, or local polynomial degree.Source: §2.1 · p. 4, §2 · p. 4
2.2. Spatial discretization using projected Kirchhoff transform.
- The semidiscrete scheme applies the same discrete Laplacian to both parts of the split and projects the nonlinear Kirchhoff transform onto the approximation space.The implicit linear solve uses a positive-definite operator, while the projection evaluates the nonlinear function at a known state.Source: §2.2 · pp. 4–5
2.3. Discrete energy functional.
- The discrete energy gradient is the projected Kirchhoff transform, and the semidiscrete energy decreases at a rate equal to the nonnegative discrete dissipation.Integrating the rate gives the corresponding space-time identity.Source: §2.3 · p. 5
- The Taylor-based energy estimate requires the diffusion coefficient to stay bounded on the pointwise segments between the states being compared.For unbounded coefficients, the bound must hold on all required segments throughout the domain.Source: §2.3 · p. 5
3. EIN time discretization and general energy stability.
This section formulates EIN stages from an IMEX-RK split and establishes energy estimates under coefficient and matrix conditions. The estimates hold for arbitrary positive time steps and do not require diffusion to have a positive lower bound.
3.1. Stage equations and elimination.
- At each stage the Kirchhoff transform is evaluated at known states, and the implicit solve uses a fixed linear operator.The assumed IMEX-DIRK structure has a lower-triangular implicit array with positive diagonals and a strictly lower-triangular explicit array with non-negative sub-diagonals.Source: §3.1 · p. 6
3.2. Energy theorem and accumulated dissipation.
- The stability proof requires only that the discrete spatial operator be self-adjoint and positive semidefinite.The auxiliary stage potentials are determined by the computed stages, with temporal-array operations acting only in stage space.Source: §3.1 · p. 7, §3.2 · p. 8
- Under the stated stabilization bounds, the energy estimates hold for arbitrary positive time steps and require no positive lower bound on the nonnegative diffusion coefficient.The diffusion coefficient is bounded above on the specified pointwise state segments; the scheme does not freeze the coefficient.Source: §3.2 · pp. 7–8
- When η > 0, the one-step estimate controls stage-potential dissipation, while the L2 increment term supplies additional numerical dissipation.The accumulated estimate is a space-time discrete counterpart of the continuous energy dissipation statement.Source: §3.2 · p. 8
4. Energy stability for first- to third-order EIN methods.
This section applies energy-stability criteria to first-, second-, and third-order EIN schemes, establishing estimates for several methods while identifying one formula that does not meet the sufficient conditions.
4.1. The EIN1 method.
- EIN1 yields one-step and accumulated energy dissipation when a0 ≥ amax and the coefficient bound holds along old-to-new state segments.For an unbounded coefficient, the bound must hold along those segments throughout each element; the energy estimate does not establish that range.Source: §4.1 · pp. 8–9
4.2. The standard EIN2 method.
- The standard EIN2 estimate gives one-step energy stability and, when summed over time, a space-time energy bound under stage-segment coefficient bounds.The theorem assumes a coefficient bound on the two adjacent pointwise stage segments.Source: §4.2 · p. 9
4.3. The EIN2-LWW method.
- EIN2-LWW gives one-step energy stability and an accumulated space-time estimate when a0 ≥ amax and the coefficient is bounded on both adjacent stage segments.The summed estimate has η = q = 1.Source: §4.3 · p. 10
4.4. The EIN3-FTY method.
- The four-stage third-order EIN3-FTY formula satisfies the energy-stability criterion when a0 ≥ amax and the coefficient is bounded on all four adjacent stage segments.Its exact rational realization is given for the scheme of [22].Source: §4.4 · p. 10
4.5. The EIN3-LWW method.
- EIN3-LWW satisfies the energy-stability criterion under the corresponding bound on all four state segments and a0 ≥ amax.All four implicit solves reuse a single operator, which may have computational benefits.Source: §4.5 · p. 11
4.6. The EIN3-ARS(4,4,3) method: Failure in the framework.
- The EIN3-ARS(4,4,3) coefficient matrices fail the sufficient conditions of Theorem 3.2, and increasing a0 cannot make them satisfy those conditions.This failure does not establish that the formula is energy unstable.Source: §4.6 · p. 12
5. Generalizations.
The analysis extends EIN energy stability to diffusion systems with a scalar energy potential in a fixed positive definite metric. It also covers consistent quadrature and additional boundary conditions under stated operator and coefficient conditions.
5.1. Diffusion systems with a scalar energy potential.
- A scalar convex potential and fixed positive definite metric H represent the system through HA(u) = ∇²Ψ(u), allowing diffusion matrices that need not be symmetric.The vector Kirchhoff transform is Φ(u) = H⁻¹(∇Ψ(u) − ∇Ψ(u∗)); for nonzero constant boundary data, the proof uses a shifted potential and transform.Source: §5.1.1 · p. 13
5.1.1. A fixed-metric potential structure.
- A fixed symmetrizer alone is insufficient: the differential compatibility condition is also needed for a scalar potential, and the stated structure is sufficient but not claimed necessary.The compatibility condition integrates each row to a vector-field component; symmetry then makes that field a gradient.Source: §5.1.1 · p. 13
5.1.2. Projected vector Kirchhoff transform and energy estimate.
- The projected vector Kirchhoff transform yields one-step and time-summed energy estimates under the theorem’s coefficient and temporal-array conditions.The fixed metric retains componentwise linear implicit solves; the estimates are stated in the H inner product.Source: §5.1.2 · pp. 14–15
5.2. Quadrature and other boundary conditions.
- Positive-weight quadrature applies to nonlinear integrals when the energy and projected transform are defined consistently; exactness for products in Vh suffices in one formulation.Alternatively, the quadrature inner product must be positive definite, with an orthogonal projection and a self-adjoint positive semidefinite discrete Laplacian; that formulation dissipates the quadrature energy.Source: §5.1.2 · p. 15, §5.2 · p. 15
- Homogeneous Neumann and periodic boundaries are covered with a corresponding self-adjoint positive semidefinite discrete Laplacian, without additional stage constraints.The estimates also extend to the fixed-metric systems.Source: §5.2 · p. 15
6. Applications.
The applications identify diffusion models that fit the energy framework, with coefficient bounds required on the relevant stage segments. Some examples have global bounds, while others require a bounded state range.
6.1. Gas flow in a porous medium.
- For segments satisfying 0 ≤ ρ ≤ R, porous-medium flow has coefficient bound amax = DmR^(m−1); in the isothermal ideal-gas case, γ = 1 and hence m = 2.Source: §6.1 · p. 16
6.2. Horizontal unsaturated flow.
- For horizontal Richards flow, the estimates require an upper bound on D over the relevant moisture range; gravity is a separate transport term.A compact range inside the constitutive interval provides such a bound when D is continuous.Source: §6.2 · p. 16
6.3. Electron heat conduction in a plasma.
- For plasma heat conduction, the convex functional is a mathematical energy for the diffusion equation and need not equal the plasma’s thermodynamic energy.The coefficient bound is required on segments with 0 ≤ T ≤ R.Source: §6.3 · p. 16
6.4. Multicomponent diffusion with constant Onsager mobility.
- The multicomponent Onsager example assumes constant symmetric positive definite mobility and restricts required state segments away from singular concentration limits.The theorem applies with a coefficient bound when each component, including the solvent fraction, stays at least δ > 0.Source: §6.4 · p. 17
6.5. Regularized vector fast diffusion.
- Regularized vector fast diffusion has a global coefficient bound Amax = εp−1, so the energy theorems need no pointwise range bound.The stated Hessian eigenvalues are positive and bounded by Amax.Source: §6.5 · p. 17
6.6. Globally bounded nonlinear cross diffusion.
- The globally bounded cross-diffusion examples use Amax = 1 + 2κ and Amax = 2 + 3κ, respectively.The second example has a generally nonsymmetric diffusion matrix but a symmetric positive definite weighted form.Source: §6.6 · p. 18
7. Concluding remarks.
The paper establishes nonlinear energy stability for first- to third-order EIN discretizations of degenerate diffusion equations. The results extend to fixed-metric systems and selected quadrature and boundary-condition settings, subject to the stated structural and coefficient bounds.
7. Concluding remarks.
- First- to third-order EIN methods dissipate a discrete convex energy under the stated coefficient bounds, without a time-step restriction or a positive lower bound on the nonnegative diffusion coefficient.Implicit stages require linear discrete backward heat solves, and the nonlinear terms are evaluated explicitly.Source: §7 · p. 18
- The spatial framework requires a self-adjoint positive semidefinite discrete operator and orthogonal projection of the Kirchhoff transform; it includes many symmetric Galerkin discretizations.Examples include CG, mixed FEM, DG variants, spectral Galerkin, and mixed VEM.Source: §7 · p. 18
- The proof extends to scalar-potential systems in a fixed positive definite metric, while some applications require bounded stage ranges to obtain a finite common coefficient bound.Regularized vector fast diffusion and bounded cross-diffusion examples admit global coefficient bounds.Source: §7 · p. 18