Numerical Methods and Applications for Hyperbolic and Parabolic Problems

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (10 May 2022) | Viewed by 9235
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Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini, 19, 00185 Rome, Italy
Interests: differential and integro-differential models; numerical schemes for hyperbolic problems; road traffic modeling; canal networks

Special Issue Information

Dear Colleagues,

Scientific computing and mathematical modeling (both deterministic and stochastic) are fundamental tools for the solution of problems arising in the analysis of complex systems.

The study of hyperbolic and parabolic problems has expanded very rapidly in recent years and finds applications—current and potential—in various fields, such as engineering, economy and finance, biomedicine and cultural heritage safeguarding, among others.

The difficulties that arise in application are both theoretical and numerical, precisely in modeling and in their numerical implementations.

This volume aims to collect high-quality articles in this vast field of research, ranging from modeling to numerical simulations.

Dr. Maya Briani
Guest Editor

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Keywords

  • hyperbolic equations
  • parabolic equations
  • conservation laws
  • numerical methods
  • applied mathematics
  • computational mathematics

Published Papers (4 papers)

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Research

27 pages, 1383 KiB  
Article
Projective Integration for Hyperbolic Shallow Water Moment Equations
by Amrita Amrita and Julian Koellermeier
Axioms 2022, 11(5), 235; https://doi.org/10.3390/axioms11050235 - 18 May 2022
Viewed by 1971
Abstract
In free surface flows, shallow water models simplify the flow conditions by assuming a constant velocity profile over the water depth. Recently developed Shallow Water Moment Equations allow for variations of the velocity profile at the expense of a more complex PDE system. [...] Read more.
In free surface flows, shallow water models simplify the flow conditions by assuming a constant velocity profile over the water depth. Recently developed Shallow Water Moment Equations allow for variations of the velocity profile at the expense of a more complex PDE system. The resulting equations can become stiff depending on the friction parameters, which leads to severe time step constraints of standard numerical schemes. In this paper, we apply Projective Integration schemes to stiff Shallow Water Moment Equations to overcome the time step constraints in the stiff regime and accelerate the numerical computations while still achieving high accuracy. In different dam break and smooth wave test cases, we obtain a speedup of up to 55 with respect to standard schemes. Full article
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19 pages, 9569 KiB  
Article
High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations
by Jianying Wei, Yongbin Ge and Yan Wang
Axioms 2022, 11(3), 111; https://doi.org/10.3390/axioms11030111 - 3 Mar 2022
Cited by 1 | Viewed by 2331
Abstract
In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. [...] Read more.
In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes. Full article
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19 pages, 755 KiB  
Article
Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations
by Yasuhiro Takei and Yoritaka Iwata
Axioms 2022, 11(1), 28; https://doi.org/10.3390/axioms11010028 - 10 Jan 2022
Cited by 5 | Viewed by 2504
Abstract
A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is [...] Read more.
A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is presented. The order of total calculation cost is O(Nlog2N). As a benchmark, the relations between numerical accuracy and discretization unit size and that between the stability of calculation and discretization unit size are demonstrated for both linear and nonlinear cases. Full article
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21 pages, 612 KiB  
Article
Reconstructed Interpolating Differential Operator Method with Arbitrary Order of Accuracy for the Hyperbolic Equation
by Shijian Lin, Qi Luo, Hongze Leng and Junqiang Song
Axioms 2021, 10(4), 295; https://doi.org/10.3390/axioms10040295 - 6 Nov 2021
Viewed by 1262
Abstract
We propose a family of multi-moment methods with arbitrary orders of accuracy for the hyperbolic equation via the reconstructed interpolating differential operator (RDO) approach. Reconstruction up to arbitrary order can be achieved on a single cell from properly allocated model variables including spatial [...] Read more.
We propose a family of multi-moment methods with arbitrary orders of accuracy for the hyperbolic equation via the reconstructed interpolating differential operator (RDO) approach. Reconstruction up to arbitrary order can be achieved on a single cell from properly allocated model variables including spatial derivatives of varying orders. Then we calculate the temporal derivatives of coefficients of the reconstructed polynomial and transform them into the temporal derivatives of the model variables. Unlike the conventional multi-moment methods which evolve different types of moments by deriving different equations, RDO can update all derivatives uniformly via a simple linear transform more efficiently. Based on difference in introducing interaction from adjacent cells, the central RDO and the upwind RDO are proposed. Both schemes enjoy high-order accuracy which is verified by Fourier analysis and numerical experiments. Full article
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