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Energy Stability and Convergence of SAV Block-centered Finite Difference Method for Gradient Flows

Xiaoli Li, Jie Shen, Hongxing Rui

arXiv:1812.01793v1math.NAcs.NA

TL;DR

Existing gradient-flow schemes face trade-offs between energy stability, nonlinear solves, and accuracy. This paper constructs a SAV/CN method with block-centered finite differences and analyzes it for gradient flows. The scheme is unconditionally energy stable and has rigorous second-order convergence in time and space under the stated assumptions.

  • Problem

    Convex splitting often yields nonlinear schemes and linear stabilization is usually first-order, while existing SAV error analysis established first-order convergence for the semi-discrete scheme.

    Source: §1 · p. 1

  • Method

    The paper constructs a SAV/CN scheme with block-centered finite differences and analyzes its stability and errors for gradient flows.

    Source: §1 · p. 2

  • Results

    The scheme is unconditionally energy stable for any time step and has rigorous second-order convergence in time and space without a uniform Lipschitz assumption on the nonlinear potential.

    Source: §1 · p. 2, §3.1 · p. 6, §3.2 · p. 7

  • Takeaways & Limitations

    The block-centered method approximates the phase function and chemical potential with Neumann boundary conditions at second-order accuracy and guarantees local mass conservation.

    Source: §1 · p. 2

    The SAV formulation assumes the energy is shifted by C0 so that E1(φ)+C0 is positive.

    Source: §2.1 · p. 3

Abstract

from arXiv · show

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

1. Introduction.

The paper develops and analyzes a SAV/CN scheme with block-centered finite differences for gradient flows. It addresses the trade-offs among energy stability, computational cost, and accuracy in existing approaches.

1. Introduction.

  • Convex splitting can yield unconditionally energy stable schemes but usually requires nonlinear solves, while linear stabilization is usually limited to first-order accuracy.Source: §1 · p. 1
  • IEQ enables linear, second-order, unconditionally energy stable schemes, but its variable-coefficient coupled systems may be difficult or expensive to solve [30].Source: §1 · p. 1
  • The SAV approach inherits key IEQ advantages while overcoming some shortcomings, and prior work established first-order convergence and error estimates for a semi-discrete SAV scheme [20].Source: §1 · p. 1
  • The proposed linear scheme has rigorous second-order convergence in time and space without assuming a uniform Lipschitz condition on the nonlinear potential.The analysis uses summation-by-parts in space and time and an induction argument to bound the numerical solution in the discrete L∞ norm.Source: §1 · p. 2

2. The SAV/CN-BCFD scheme.

The scheme combines a scalar auxiliary variable Crank–Nicolson method with block-centered finite differences for gradient flows. It is described for both H−1 and L2 flows.

2.1. The semi discrete SAV/CN scheme.

  • The semi-discrete SAV/CN scheme shifts E1 by C0 so that the energy is positive, then applies the Crank–Nicolson discretization.The shift is chosen with C0 > c0, where the energy density integral is bounded below by −c0.Source: §2.1 · p. 3

2.2. Spacial discretization.

  • The spatial discretization uses block-centered finite differences on staggered grids and discrete inner products and norms.Source: §2.2 · pp. 3–4

2.2.1. SAV/CV-BCFD scheme for H−1 gradient flow.

  • For the H−1 gradient flow, the paper specifies a SAV/CV-BCFD scheme and its boundary and initial approximations.Source: §2.2.1 · pp. 4–5

2.2.2. SAV/CV-BCFD scheme for L2 gradient flow.

  • For the L2 gradient flow, the paper likewise specifies a SAV/CV-BCFD scheme, with the boundary and initial conditions given by (2.16).Source: §2.2.2 · p. 5

3. Unconditional energy stability.

The section establishes unconditional discrete energy stability for the full SAV/CN-BCFD schemes, with energy laws for both H−1 and L2 gradient flows.

3. Unconditional energy stability.

  • The full SAV/CN-BCFD schemes are unconditionally energy stable with a discrete energy functional.Source: §3 · p. 5

3.1. H−1 gradient flow.

  • For H−1 gradient flow, the scheme satisfies a discrete energy law for any ∆t.Source: §3.1 · p. 6

3.2. L2 gradient flow.

  • For L2 gradient flow, the scheme is unconditionally stable and satisfies a discrete energy law for any ∆t.Source: §3.2 · p. 7

4. Error estimates.

The section presents error estimates for the fully discrete SAV/CN-BCFD schemes, treating H−1 and L2 gradient flows.

4. Error estimates.

  • The section derives error estimates for the fully discrete SAV/CN-BCFD schemes.Source: §4 · p. 7

4.1. H−1 gradient flow.

  • The H−1 error-estimate theorem assumes solution regularity and requires ∆t ≤ C(hx + hy).Source: §4.1 · p. 7

4.2. L2 gradient flow.

  • The L2 error-estimate theorem assumes solution regularity and requires ∆t ≤ C(hx + hy).Source: §4.2 · p. 16

5. Numerical simulations.

Numerical experiments test SAV/CN-BCFD accuracy for Allen–Cahn and Cahn–Hilliard equations and examine adaptive stepping in coarsening dynamics. They report second-order convergence and adaptive solutions consistent with the small-step reference, while large-step solutions deviate.

5.1.1. Convergence rates of the SAV/CN-BCFD scheme for Allen-Cahn equation.

  • The Allen–Cahn example exhibits the predicted second-order convergence in the reported discrete-norm errors.The errors and convergence rates are listed in Table 1.Source: §5.1.1 · p. 17, Table 1 · p. 17

5.1.2. Convergence rates of SAV/CN-BCFD scheme for Cahn-Hilliard equation.

5.2. Coarsening dynamics and adaptive time stepping.

  • Because the scheme is unconditionally energy stable, adaptive time steps can be chosen according to accuracy rather than stability.The strategy adjusts step sizes to accommodate intervals where energy and solution change at different rates.Source: §5.2 · pp. 17–18
  • Adaptive-step solutions agree with the small-step reference, whereas large-step solutions deviate; adaptive steps vary by about two orders of magnitude.The comparisons include phase snapshots, discrete scaled surface energy, and roughness.Source: §5.2 · pp. 18–19, Figure 1 · p. 19, Figure 2 · p. 19
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