Schwartz duality for singularly perturbed nonlinear differential equations with Chebyshev spectral method

TL;DR

Dirac-delta sources in singular perturbations induce Gibbs oscillations in high-order spectral methods, especially for nonlinear convection. The authors extend Schwartz duality to nonlinear problems by representing the Dirac delta as a scaled distribution using the local solution within Chebyshev collocation, yielding a consistent discretization. The approach eliminates Gibbs oscillations across 1D and 2D nonlinear problems, including multiple point sources and higher-degree nonlinearities, and is compatible with both spectral and finite-difference collocation schemes. This enables accurate, non-oscillatory solutions for nonlinear PDEs with singular sources across dimensions, with broad potential for practical numerical methods in applied settings.

Abstract

Singularly perturbed differential equations with a Dirac delta function yield discontinuous solutions. Therefore, careful consideration is required when using numerical methods to solve these equations because of the Gibbs phenomenon. A remedy based on the Schwartz duality has been proposed, yielding superior results without oscillations. However, this approach has been limited to linear problems and still suffers from the Gibbs phenomenon for nonlinear problems. In this note, we propose a consistent yet simple approach based on Schwartz duality that can handle nonlinear problems. Our proposed approach utilizes a modified direct projection method with a discrete derivative of the Heaviside function, which directly approximates the Dirac delta function. This proposed method effectively eliminates Gibbs oscillations without the need for traditional regularization and demonstrates uniform error reduction.

Figures (5)

• Figure 1: $(\mathbf{u} - \vec{\beta}) / (\alpha - \beta)$ versus $x$, where $\mathbf{u}$ represents the numerical solution of Eq. \ref{['eq:time_dep_2']} with $c = -0.1$, $\beta = 2$, $C=0.5$ and $N=64$. Left: The previous Schwartz method at time $t= 1.63$. Right: The proposed method at time $t= 1.63$. The exact $\mathbf{H}_c$ is given in blue solid line and the approximation is shown with red dots.
• Figure 2: $|\mathbf{H}_c - (\mathbf{u}-\vec{\beta})/(\alpha - \beta)|$ for various degrees $m = 2, 3, \ldots, 10$, for the previous method (Left) and the proposed method (Right) with $c = -0.1$, $\beta = 2$, $C=0.5$ and $N=64$ of the numerical method (\ref{['eq:proposed_general_numerical']}).
• Figure 3: Numerical simulation $\mathbf{u}$ of Eq. \ref{['eq:burgers multiple source']} with $N=256$, $\beta=2$, $C=0.5$ and the source term $0.5\delta(x+0.7) + 2\delta(x) + \delta(x-0.3)$ by Gaussian approximation with the standard derivation $\sigma=0.02$ (left, green) and the proposed method (right, red) at time $t=1.5$. The exact solution $2 + (\sqrt{5}-2)H_{-0.7}+ (3-\sqrt{5})H_{0} + (\sqrt{11}-3)H_{0.3}$ is given in blue solid line.
• Figure 4: Numerical solutions of Eqs. \ref{['eq:2d_advec']} with $a=b=1$ and \ref{['eq:2d-nonlinear']} by approximating the delta function $\delta_{(0,0)}$ as $\mathbf{T}D^t+D\mathbf{T}$ in (a) and (b), and as $\frac{2\mathbf{u}}{\mathbf{u}(x_i^*,x_i^*)+\mathbf{u}(x_{i+1}^*,x_{i+1}^*)}(\mathbf{T}D^t+D\mathbf{T})$ in (c) with $\beta = 2$, $C=0.5$ and $N=64$ at time $t= 1$.
• Figure 5: Numerical solution of Eqs. \ref{['eq:2d_advec']}with $a=b=1$ and \ref{['eq:2d-nonlinear']} by approximating the delta function $\delta_{(0,0)}$ as two-dimensional Gaussian functions at time $t=2$ with $\beta = 2$, $N=128$ and the standard deviation $\sigma=0.1$ for figures (a) and (b). For Figure (c), the delta function $\delta_{(0,0)}$ is approximated $\frac{2\mathbf{u}}{\mathbf{u}(x_i^*,x_i^*)+\mathbf{u}(x_{i+1}^*,x_{i+1}^*)}(\mathbf{T}D^t+D\mathbf{T})$ at time $t=1$ with $\beta = 2$, $C=0.5$ and $N=64$. Here $D$ is the forward difference matrix.

Theorems & Definitions (6)

• Proposition 3.1
• proof
• Proposition 4.1
• proof
• Theorem 4.2
• proof