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Related Concept Videos

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Linear time-invariant Systems01:23

Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
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First Order Systems01:21

First Order Systems

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Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the wave equation, two different graphs can be plotted.
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Equations of Wave Motion

Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.

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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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Inf-Sup Stable Space-Time Discretization of the Wave Equation Based on a First-Order-In-Time Variational Formulation.

Matteo Ferrari1, Ilaria Perugia1, Enrico Zampa1

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Journal of Scientific Computing
|May 7, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a new space-time method for the wave equation, ensuring stability for various discrete subspaces. The approach achieves optimal convergence rates, validated by numerical examples.

Keywords:
First-order-in-time formulationInf–sup stabilitySpace–time methodWave equation

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Partial Differential Equations

Background:

  • The wave equation is fundamental in physics and engineering.
  • Existing discretization methods often face stability limitations or mesh restrictions.
  • A robust and flexible numerical method is needed for accurate wave propagation simulations.

Purpose of the Study:

  • To develop a conforming space-time discretization for the first-order wave equation.
  • To extend existing methods by incorporating exponential time weights for enhanced stability.
  • To achieve optimal convergence rates without mesh or time-step restrictions.

Main Methods:

  • Utilizing a first-order-in-time variational formulation.
  • Incorporating exponential weights in the time discretization.
  • Applying elliptic projections to derive convergence properties.
  • Employing conforming space-time tensor product subspaces.

Main Results:

  • Established an inf-sup stability condition for arbitrary discrete subspaces, including splines.
  • Achieved optimal convergence rates in energy and L^2 norms for smooth solutions.
  • Demonstrated stability independent of mesh size and time step.
  • Numerical examples confirmed the theoretical convergence rates.

Conclusions:

  • The proposed space-time discretization offers a stable and accurate numerical solution for the wave equation.
  • The method's flexibility in choosing discrete subspaces and its independence from mesh/time-step constraints make it broadly applicable.
  • This work provides a robust foundation for simulating wave phenomena with high fidelity.