Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

Huang, Yu; Skandari, M. H. Noori; Mohammadizadeh, Fatemeh; Tehrani, H. A.; Georgiev, Svetlin Georgiev; Tohidi, Emran; Shateyi, Stanford

doi:10.3390/sym11121439

Cited by 7 publications

(2 citation statements)

References 38 publications

(67 reference statements)

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“…This is why in practical calculations often small artificial damping is added to the model. (3) Spatial-temporal semianalytical basis function methods [37][38][39] employ the spatial-temporal semi-analytical basis function a priori to satisfy the transient wave equation and then solve it directly. Among these three time-discretization schemes, the first two have been widely used for transient wave propagation analysis; the last one has not been widely used because the time-dependent semi-analytical basis functions are not easy to construct, in particular, the transient wave equation, including the source excitations.…”

Section: Introductionmentioning

confidence: 99%

A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation

Chen

2022

Mathematics

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In this paper, a novel semi-analytical collocation solver, the spatial–temporal radial Trefftz collocation method (STRTCM) is proposed to solve 3D transient wave equations with specified sound source excitations. Unlike the traditional time discretization strategies, the proposed numerical scheme introduces the spatial–temporal radial Trefftz functions (STRTFs) as the basis functions for the spatial and temporal discretization of the transient wave equations. The STRTFs are constructed in the spatial–temporal domain, which is a combination of 3D Euclidean space and time into a 4D manifold. Moreover, since the initial and boundary conditions are imposed on the spatial–temporal domain boundaries, the original transient wave propagation problem can be converted to an inverse boundary value problem. To deal with the specified time-dependent sound source excitations, the composite multiple reciprocity technique is extended from the spatial domain to the spatial–temporal domain, which transforms the original problem with a source term into a high-order problem without a source term. By deriving the related STRTFs for the considered high-order problem, the proposed scheme only requires the node discretization on the spatial–temporal domain boundaries. The efficiency of the proposed method is numerically verified by four benchmark examples under 3D transient wave equations with specified time-dependent sound source excitation.

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Section: Introductionmentioning

confidence: 99%

A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation

Chen

2022

Mathematics

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“…We fully show that the approximate solutions are convergent to the exact solution when the number of collocation points tends to infinity. Note that spectral collocation methods have high accuracy and exponential convergence and, up to now many researchers utilized them to solve different continuous-time problems involving the ordinary and partial differential equations [12,13,18].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Reaction–Diffusion Equations with Convergence Analysis

Heidari

Ghovatmand²,

Skandari³

et al. 2022

J Nonlinear Math Phys

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In this manuscript, we implement a spectral collocation method to find the solution of the reaction–diffusion equation with some initial and boundary conditions. We approximate the solution of equation by using a two-dimensional interpolating polynomial dependent to the Legendre–Gauss–Lobatto collocation points. We fully show that the achieved approximate solutions are convergent to the exact solution when the number of collocation points increases. We demonstrate the capability and efficiency of the method by providing four numerical examples and comparing them with other available methods.

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Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

Cited by 7 publications

References 38 publications

A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation

A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation

Numerical Solution of Reaction–Diffusion Equations with Convergence Analysis

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