[Paper Review] Finite difference approximations for the first-order hyperbolic partial differential equation with point-wise delay

Paramjeet Singh, Kapil K. Sharma|arXiv (Cornell University)|Dec 5, 2010

Stability and Controllability of Differential Equations23 references3 citations

TL;DR

This paper proposes second-order accurate finite difference schemes—Lax-Friedrichs and Leap-Frog— for solving first-order hyperbolic PDEs with point-wise delay, ensuring consistency, stability under the CFL condition, and convergence. The method handles both small and large delays by constructing a specialized grid where delayed arguments align with grid points, validated through 1D and 2D numerical examples showing second-order convergence in space and time.

ABSTRACT

Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance, i.e., shift in space. The differential equation involving point-wise delay and advance models the distribution of the time intervals between successive neuronal firings. We construct higher order numerical approximations and discuss their consistency, stability and convergence. The numerical approximations constructed in this paper are consistent, stable under CFL condition, and convergent. We also extend our methods to the higher space dimensions. Some test examples are included to illustrate our approach. These examples verify the theoretical estimates and shows the effect of point-wise delay on the solution.

Motivation & Objective

• To develop higher-order numerical methods for first-order hyperbolic PDEs with point-wise delay or advance, extending prior first-order schemes.
• To ensure numerical stability and convergence under the CFL condition for problems with non-constant, smooth coefficients.
• To construct a specialized finite difference mesh that aligns delayed spatial arguments with grid points, enabling accurate approximation.
• To extend the method to two and higher space dimensions, maintaining second-order accuracy.
• To demonstrate the effect of point-wise delay on solution behavior through numerical test cases.

Proposed method

• The Lax-Friedrichs and Leap-Frog finite difference schemes are applied to the hyperbolic PDE with point-wise delay, using a staggered grid that includes the delayed spatial domain $[-\alpha, 0]$.
• A special mesh is constructed so that the delayed argument $x - \alpha$ maps exactly to a grid point, ensuring the discrete solution can access delayed values directly.
• The schemes are derived by replacing time and space derivatives with second-order accurate finite difference approximations, preserving consistency.
• Stability is analyzed under the CFL condition, which restricts the time step relative to spatial step to maintain numerical stability.
• The method is extended to two spatial dimensions by applying the same finite difference stencils in both $x$ and $y$ directions, with boundary conditions defined on $[-\alpha, 0] \times [-\beta, 0]$.
• Numerical solutions are computed iteratively using initial and boundary data, with delayed values retrieved from stored grid points.

Experimental results

Research questions

• RQ1 Can second-order accurate finite difference schemes be constructed for first-order hyperbolic PDEs with point-wise delay, ensuring consistency and convergence?
• RQ2 How does the CFL condition affect the stability of explicit schemes like Lax-Friedrichs and Leap-Frog in the presence of delay?
• RQ3 To what extent does the point-wise delay alter the solution profile, and can this be captured accurately by the proposed numerical method?
• RQ4 Can the method be extended to two and higher spatial dimensions while preserving second-order accuracy and stability?
• RQ5 How do the numerical errors scale with mesh refinement, and does the method achieve second-order convergence in space and time?

Key findings

• The proposed Lax-Friedrichs and Leap-Frog schemes are consistent, unconditionally stable under the CFL condition, and convergent for the hyperbolic PDE with point-wise delay.
• Numerical experiments confirm second-order convergence in space and time, as evidenced by error tables showing error reduction by a factor of approximately 4 when grid spacing is halved.
• For Example 1 with $\alpha = 0.05$, the maximum absolute error for the Lax-Friedrichs scheme decreased from 0.053623 ($\Delta x = 1/100$) to 0.004520 ($\Delta x = 1/800$), indicating second-order convergence.
• For Example 2 with $\alpha = 0.1$, the Leap-Frog scheme showed a similar reduction in root mean square error, from 0.041308 to 0.000612 over the same refinement sequence.
• The 2D numerical experiment with $\alpha = \beta = 0.5$ confirmed the method’s applicability in higher dimensions, with stable and accurate solutions visualized at $t = 0.5$.
• The effect of delay on solution behavior is clearly visible in plots, showing phase shifts and amplitude modulation consistent with theoretical expectations.