[Paper Review] Residual equilibrium schemes for time dependent partial differential equations

Lorenzo Pareschi, Thomas Rey|arXiv (Cornell University)|Feb 8, 2016

Computational Fluid Dynamics and Aerodynamics38 references19 citations

TL;DR

This paper introduces residual equilibrium schemes for time-dependent PDEs that preserve steady-state solutions while achieving high-order accuracy. By modifying numerical fluxes using equilibrium-based corrections and flux limiters, the method ensures well-balanced behavior for convection-diffusion, hyperbolic balance laws, and kinetic equations, maintaining stability and monotonicity without sacrificing accuracy.

ABSTRACT

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.

Motivation & Objective

• To develop numerical schemes that preserve nontrivial steady-state solutions in time-dependent PDEs, especially when standard methods fail due to source-term balance.
• To extend micro-macro decomposition techniques from kinetic equations to a broader class of PDEs, including nonlinear diffusion and hyperbolic balance laws.
• To achieve high-order accuracy while maintaining well-balanced and TVD (total variation diminishing) properties in the presence of source terms.
• To provide a structurally simple, widely applicable framework that avoids problem-specific flux construction for steady-state preservation.

Proposed method

• The method introduces a residual distribution approach where numerical fluxes are corrected by subtracting equilibrium fluxes derived from the steady-state solution.
• A flux limiter function φ(r) is introduced, depending on an equilibrium indicator r = ΔUⁿᵢ / ΔUᵉᑫᵢ, to control the correction and preserve monotonicity.
• The scheme is formulated as ∂uᵢ/∂t = −(𝒰ᵢ₊₁/₂ − 𝒰ᵢ₋₁/₂)/Δx + (uᵢ − uᵢᵉᑫ), where 𝒰ᵢ±₁/₂ = Uᵘᵐᵢ±₁/₂ − φᵢUᵉᑫᵢ±₁/₂, ensuring steady-state preservation.
• For linear problems, the transformation uᵢ → uᵢ − uᵢᵉᑫ makes the residual scheme equivalent to the underlying method, preserving its properties.
• For nonlinear schemes, the flux limiter φ(r) is designed such that 0 ≤ φ(r)/r ≤ 1, ensuring TVD stability under appropriate time-step conditions.
• The approach is validated on convection-diffusion equations, hyperbolic systems with source terms, and kinetic equations, with numerical results confirming accuracy and well-balanced behavior.

Experimental results

Research questions

• RQ1 Can a general framework be developed to preserve steady-state solutions in time-dependent PDEs without requiring problem-specific flux design?
• RQ2 How can high-order accuracy be maintained in well-balanced schemes for nonlinear PDEs with source terms?
• RQ3 What flux limiter strategy ensures both steady-state preservation and TVD (total variation diminishing) stability in nonlinear schemes?
• RQ4 To what extent can micro-macro decomposition techniques be generalized beyond kinetic equations to other PDE classes?

Key findings

• The residual equilibrium scheme exactly preserves the steady-state solution uᵉᑫ(𝑥) for convection-diffusion equations, as demonstrated by the semi-discrete form ∂uᵢ/∂t = −(𝒰ᵢ₊₁/₂ − 𝒰ᵢ₋₁/₂)/Δx + (uᵢ − uᵢᵉᑫ).
• For the linear convection-diffusion problem with uᵉᑫ(𝑥) = u_B exp(−𝑥/a), the method maintains the steady state regardless of mesh resolution or time step, provided the flux correction is applied.
• The flux limiter φ(r) defined as r^α for r ≤ 1 and r^−α for r > 1 (with α > 1) ensures both well-balanced behavior and TVD stability under the condition 0 ≤ φ(r)/r ≤ 1.
• Numerical results confirm that the scheme maintains high-order accuracy and avoids spurious oscillations, even in the presence of strong source terms and discontinuous initial data.
• The method preserves the TVD property of the underlying scheme, as shown by the inequality TV(uⁿ⁺¹) ≤ (1 − Δt)TV(uⁿ) + ΔtTV(uᵉᑫ), provided TV(uᵉᑫ) ≤ TV(uⁿ).
• The approach is extendable to systems of conservation laws with source terms and to kinetic equations, demonstrating broad applicability beyond the prototype problems considered.