Properties of a Fourth-Order Vector Compact Scheme for Acoustic Wave Equation

Core Concepts

Stability and error bounds of a semi-explicit fourth-order vector compact scheme for the multidimensional acoustic wave equation are proven.

Abstract

The content discusses a three-level semi-explicit in time higher-order vector compact scheme for the multidimensional acoustic wave equation. Stability and 4th order error bound in an enlarged energy norm are derived. Results from 3D numerical experiments show high accuracy for smooth data, outperforming explicit 2nd order schemes for nonsmooth data. The scheme's implementation does not require iterations, demonstrating advantages over implicit schemes. Various stability theorems and error bounds are discussed, emphasizing conditional stability inherent to compact 4th order schemes.

Introduction:

Study continuation on semi-explicit fourth-order vector compact scheme.
Use of additional sought functions approximating spatial derivatives.

Acoustic Wave Equation and Scheme:

Initial-boundary value problem formulation.
Three-level semi-explicit in time vector compact scheme details.

Stability Analysis:

Theorem on stability in enlarged energy norm derived.
Corresponding 4th order error bound established.

Numerical Experiments:

Conducted on various tests with smooth and nonsmooth data.
Convergence rates analyzed based on mesh parameters N and M.

Error Bounds:

Error analysis shows excellent results with convergence rates close to theoretical values.

Practical Implementation:

Code written in C language for x64 architecture with detailed computational specifications provided.

Stats

For a variable speed of sound, ρ(x)∂2t u(x, t) −Lu(x, t) = f(x, t), where L := a21∂21 + ... + a2n∂2n.
Stability theorem: max(ε0∥vm∥Bh, ∥Imht¯stv∥Ah) ⩽ ∥v0∥Bh + 2∥A−1/2hu(1)∥h + 2A−1/2hφ˜L1ht(Hh).
Error bound: √εε0max(ρ1/2‖¯δtrm‖h, ‖rm‖Eh, ‖ImhtEhr‖Iρh) = O(|h|4).

Quotes

Key Insights Distilled From

On properties of a semi-explicit in time fourth-order vector compact scheme for the multidimensional acoustic wave equation

by Alexander Zl... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.16174.pdf

Deeper Inquiries

How does the proposed scheme compare to other existing methods

The proposed scheme in the study offers a semi-explicit fourth-order vector compact scheme for solving the multidimensional acoustic wave equation. This scheme has several advantages compared to other existing methods. Firstly, it demonstrates stability and derives a 4th order error bound in an enlarged energy norm, showcasing its accuracy and efficiency. Additionally, the scheme does not require iterations for implementation, simplifying the computational process. The use of additional sought functions that approximate 2nd order non-mixed spatial derivatives contributes to its high accuracy for smooth data and improved error behavior over classical explicit 2nd order schemes for nonsmooth data.

What challenges might arise when applying this scheme to real-world acoustic wave problems

When applying this scheme to real-world acoustic wave problems, several challenges may arise. One challenge is ensuring that the assumptions made in the theoretical analysis hold true in practical scenarios. Real-world data may not always align perfectly with the idealized conditions assumed in mathematical models, leading to potential discrepancies between expected and observed results. Another challenge is scalability - while the scheme shows promise in numerical experiments on smaller scales, implementing it effectively for larger or more complex systems may require significant computational resources and optimization techniques.

How can the findings of this study be extended to address more complex wave equations

The findings of this study can be extended to address more complex wave equations by exploring different boundary conditions, variable coefficients, or nonlinear terms within the equations. By adapting the semi-explicit fourth-order vector compact scheme to accommodate these variations, researchers can investigate how well it performs under diverse conditions and gain insights into its applicability across a wider range of scenarios. Additionally, further research could focus on validating the scheme through experimental testing or comparing it with other advanced numerical methods to assess its robustness and versatility in handling complex wave phenomena.