Front Tracking for Hyperbolic Conservation Laws: A Comprehensive Guide to Applied Mathematical Sciences

Hyperbolic conservation laws are a class of partial differential equations that describe the conservation of physical quantities such as mass, momentum, and energy. These equations are widely used in a variety of applications, including fluid dynamics, gas dynamics, and elasticity.

One of the main challenges in solving hyperbolic conservation laws is the presence of sharp interfaces, such as shock waves and contact discontinuities. These interfaces can introduce significant numerical errors if they are not handled properly.

Front tracking is a numerical method that is specifically designed to handle sharp interfaces. This method tracks the interfaces explicitly, and it uses a different numerical scheme for each side of the interface. This approach allows for a much more accurate representation of the solution than traditional methods that do not track the interfaces.

Front Tracking for Hyperbolic Conservation Laws (Applied Mathematical Sciences Book 152)

by Helge Holden

4.4 out of 5

Language	:	English
File size	:	22726 KB
Screen Reader	:	Supported
Print length	:	531 pages

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One of the key concepts in the theory of conservation laws is the notion of a weak solution. A weak solution is a function that satisfies the conservation law in an integral sense, but it does not necessarily satisfy the conservation law pointwise.

Front tracking methods are based on the idea of constructing weak solutions to hyperbolic conservation laws. These weak solutions are constructed by tracking the sharp interfaces explicitly and by using a different numerical scheme for each side of the interface.

Once the sharp interfaces have been identified, the next step is to construct a numerical scheme for each side of the interface. The choice of numerical scheme will depend on the specific problem being solved.

Finally, the numerical scheme is used to evolve the solution in time. The evolution process is typically carried out using a time-stepping algorithm, such as the Runge-Kutta method.

• Shock wave propagation: Front tracking methods are well-suited for solving problems involving shock wave propagation. These methods can accurately capture the shock wave and its interactions with other features in the flow.
• Contact discontinuity propagation: Front tracking methods can also be used to solve problems involving contact discontinuity propagation. These methods can accurately capture the contact discontinuity and its interactions with other features in the flow.
• Multiphase flow: Front tracking methods are well-suited for solving problems involving multiphase flow. These methods can accurately capture the interfaces between different phases and their interactions with each other.
• Elasticity: Front tracking methods can be used to solve problems in elasticity. These methods can accurately capture the deformation of elastic materials and their interactions with other objects.

Front tracking methods are a powerful numerical method for solving hyperbolic conservation laws. These methods are particularly well-suited for problems involving sharp interfaces, such as shock waves and contact discontinuities. Front tracking methods have been successfully applied to a wide variety of problems in fluid dynamics, gas dynamics, and elasticity.

Front Tracking for Hyperbolic Conservation Laws (Applied Mathematical Sciences Book 152)

by Helge Holden

4.4 out of 5

Language	:	English
File size	:	22726 KB
Screen Reader	:	Supported
Print length	:	531 pages

FREE