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Projective integration schemes for nonlinear degenerate parabolic systems

Tommaso Tenna

TL;DR

The paper addresses stiff BGK-based discretizations of degenerate parabolic systems and proposes a fully explicit, high-order scheme using projective integration to bypass restrictive inner-step CFL conditions. By coupling a few inner explicit steps with an outer projective extrapolation, and leveraging Runge–Kutta extensions for higher order, the method achieves stability and consistency, supported by spectral analysis and a Chapman–Enskog-based consistency framework. Numerical experiments in 1D and 2D across linear diffusion, advection–diffusion, Burgers-type, and Buckley–Leverett models (including degenerate diffusion and gravity) demonstrate accurate convergence, robust handling of degeneracy, and significant speedups over direct explicit schemes, with competitive performance against IMEX approaches. The work highlights the method’s asymptotic-preserving character as the relaxation parameter $oldsymbol{ε} o 0$, and its potential connections to lattice Boltzmann methods, suggesting broad applicability to multi-phase flow and related kinetic-inspired PDEs.

Abstract

A general high-order fully explicit scheme based on projective integration methods is here presented to solve systems of degenerate parabolic equations in general dimensions. The method is based on a BGK approximation of the advection-diffusion equation, where we introduce projective integration method as time integrator to deal with the stiff relaxation term. This approach exploits the clear gap in the eigenvalues spectrum of the kinetic equation, taking into account a sequence of small time steps to damp out the stiff components, followed by an extrapolation step over a large time interval. The time step restriction on the projective step is similar to the CFL condition for advection-diffusion equations. In this paper we discuss the stability and the consistency of the method, presenting some numerical simulations in one and two spatial dimensions.

Projective integration schemes for nonlinear degenerate parabolic systems

TL;DR

The paper addresses stiff BGK-based discretizations of degenerate parabolic systems and proposes a fully explicit, high-order scheme using projective integration to bypass restrictive inner-step CFL conditions. By coupling a few inner explicit steps with an outer projective extrapolation, and leveraging Runge–Kutta extensions for higher order, the method achieves stability and consistency, supported by spectral analysis and a Chapman–Enskog-based consistency framework. Numerical experiments in 1D and 2D across linear diffusion, advection–diffusion, Burgers-type, and Buckley–Leverett models (including degenerate diffusion and gravity) demonstrate accurate convergence, robust handling of degeneracy, and significant speedups over direct explicit schemes, with competitive performance against IMEX approaches. The work highlights the method’s asymptotic-preserving character as the relaxation parameter , and its potential connections to lattice Boltzmann methods, suggesting broad applicability to multi-phase flow and related kinetic-inspired PDEs.

Abstract

A general high-order fully explicit scheme based on projective integration methods is here presented to solve systems of degenerate parabolic equations in general dimensions. The method is based on a BGK approximation of the advection-diffusion equation, where we introduce projective integration method as time integrator to deal with the stiff relaxation term. This approach exploits the clear gap in the eigenvalues spectrum of the kinetic equation, taking into account a sequence of small time steps to damp out the stiff components, followed by an extrapolation step over a large time interval. The time step restriction on the projective step is similar to the CFL condition for advection-diffusion equations. In this paper we discuss the stability and the consistency of the method, presenting some numerical simulations in one and two spatial dimensions.

Paper Structure

This paper contains 27 sections, 3 theorems, 138 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Relaxation schemes for nonlinear degenerate parabolic systems
  3. Chapman-Enskog expansion
  4. Discrete kinetic models
  5. Diagonal Relaxation Model 1 (DRM1).
  6. Diagonal Relaxation Model 2 (DRM2)
  7. A 3 velocities model (OVM)
  8. Numerical Approximation
  9. Velocity discretization
  10. Spatial discretization
  11. Projective Integration
  12. Stability analysis
  13. Consistency analysis
  14. Numerical Simulations
  15. Order of convergence: linear diffusion equation
  16. ...and 12 more sections

Key Result

Theorem 3.1

The spectrum of the matrix $\mathcal{A}$ satisfies where $C$ is a constant depending on the entries of $D$, $\xi$ and $\gamma$ are defined in D_matrix_spectrum and $\zeta(\varepsilon)$ is the dominant eigenvalue.

Figures (18)

  • Figure 1: Idea of Projective Integration method.
  • Figure 2: Idea of the Projective Runge-Kutta Integration method. The scheme first computes $f^{n,K+1}$
  • Figure 3: Spectrum of the operator for $\varepsilon=10^{-7}$.
  • Figure 4: Order of convergence for the $3$rd order scheme (on the left) and the $4$th order scheme (on the right) for different norms, using the linear diffusion equation \ref{['linear_diffusion_test']}.
  • Figure 5: Order of convergence for the PRK3 (on the left) and the PRK4 (on the right) for different norms, using the linear diffusion equation \ref{['linear_diffusion_test']}.
  • ...and 13 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem C.1: lafittelejon2016, Theorem 3.2