Unconditional Stability for Numerical Scheme Combining Implicit Timestepping for Local Effects and Explicit Timestepping for Nonlocal Effects

TL;DR

Addresses unconditional stability for a mixed implicit-explicit timestepping scheme for $u'(t) + A u(t) + B(u)u(t) - C u(t) = f(t)$, where $A$ is SPD, $B(u)$ skew-symmetric and $C$ nonlocal. The method uses an implicit treatment of $A$ and $B(u)$ and an explicit treatment of $C$, and the authors prove unconditional stability via both an algebraic argument and an energy-method bound with the energy norm $\|u\|_E^2 = u^T u + k u^T C u$; a key step bounds $\|(I+kC)^{1/2}(I+kA+kB_n)^{-1}(I+kC)^{1/2}\|_2 \le 1$. They also establish $O(k)$ local truncation error and derive energy-norm convergence bounds for the nonhomogeneous problem, including a separate case $\Gamma_E=0$ when $B(u)$ is constant. Numerical experiments on turbulence-inspired discretizations confirm stability for large $k$ and illustrate the scheme's sparsity and robustness compared to fully explicit/skew schemes.

Abstract

A combination of implicit and explicit timestepping is analyzed for a system of ODEs motivated by ones arising from spatial discretizations of evolutionary partial differential equations. Loosely speaking, the method we consider is implicit in local and stabilizing terms in the underlying PDE and explicit in nonlocal and unstabilizing terms. Unconditional stability and convergence of the numerical scheme are proven by the energy method and by algebraic techniques. This stability result is surprising because usually when different methods are combined, the stability properties of the least stable method plays a determining role in the combination.

Figures (4)

• Figure 1: Spatial stability of the steady-state solution for various choices of the artificial viscosity parameter $\epsilon_0$.
• Figure 2: Stability of the numerical method demonstrated by the behavior of the energy norm
• Figure 3: Numerical validation of Theorem \ref{['t:unconditional']}
• Figure 4: Exponential growth of the solution of the scheme that includes the advection term explicitly

Theorems & Definitions (11)

• Remark 1.1
• Lemma 2.1
• Lemma 2.2
• Definition 2.1
• Theorem 2.1
• Lemma 2.3
• Theorem 2.2
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6