Related Experiment Video
Updated: Mar 7, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
9.1K
Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations
Olivier Sarbach1, Manuel Tiglio2
1Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Ciudad Universitaria, 58040 Morelia, Michoacán Mexico.
Living Reviews in Relativity
|February 10, 2017
Summary
This study reviews the theory for solving partial differential equations in physics, focusing on numerical relativity and Einstein
Area of Science:
- Physics
- Computational Science
- Astrophysics
Background:
- Evolution problems in physics often involve partial differential equations on infinite domains.
- Numerical solutions are required when analytical solutions are intractable, especially for complex systems like binary black holes.
Purpose of the Study:
- To review the theory of continuum and discrete initial-boundary value problems for hyperbolic partial differential equations.
- To discuss applications in numerical relativity, including well-posed formulations of Einstein's equations.
Main Methods:
- Discretization of infinite domains into finite computational grids.
- Development of multi-domain high-order finite difference and spectral methods.
- Analysis of well-posedness for initial and initial-boundary value problems.
Main Results:
- Established theoretical foundations for approximating solutions to hyperbolic partial differential equations.
- Presented well-posed formulations of Einstein's equations for numerical relativity.
- Demonstrated the application of advanced numerical methods for solving these problems.
Conclusions:
- Accurate approximation of solutions to hyperbolic PDEs is achievable through careful discretization and advanced numerical methods.
- The presented methods are crucial for advancing numerical relativity and understanding phenomena like binary black hole mergers.
Related Concept Videos
01:16Electrostatic Boundary Conditions
1.0K
Consider an external electric field propagating through a homogeneous medium. When the electric field crosses the surface boundary of the medium, it undergoes a discontinuity. The electric field can be resolved into normal and tangential components. The amount by which the field changes at any boundary is given by the difference between the field components above and below the surface boundary.
The surface integral of an electric field is given by Gauss's law in integral form and is related to...
The surface integral of an electric field is given by Gauss's law in integral form and is related to...
1.0K
01:27Electrostatic Boundary Conditions in Dielectrics
2.0K
When an electric field passes from one homogeneous medium to another, crossing the boundary between the two mediums imparts a discontinuity in the electric field. This results in electrostatic boundary conditions that depend on the type of mediums the field propagates through.
Consider a case where both the mediums across a boundary are two different dielectric materials. Recall that the electric field and electric displacement are proportional and related through the material's permittivity....
Consider a case where both the mediums across a boundary are two different dielectric materials. Recall that the electric field and electric displacement are proportional and related through the material's permittivity....
2.0K
01:28Magnetostatic Boundary Conditions
1.7K
An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...
1.7K
01:20Continuity Equation
1.6K
The total amount of current flowing per unit cross-sectional area is called the current density. Hence, the current passing through a cross-sectional area can be written as the surface integral of the current density.
1.6K
01:28Continuity Equation
3.5K
The continuity equation asserts that the mass flow rate must remain constant for a steady flow of an incompressible fluid within a confined system. This principle applies to systems where fluid passes through varying cross-sectional areas, such as nozzles, syringes, and pipes.
The mass flow rate is expressed as:
The mass flow rate is expressed as:
3.5K
01:28Improper Integrals: Discontinuous Integrands
72
Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
72