Chebyshev multidomain pseudospectral method to solve cardiac wave equations with rotational anisotropy

2018 ◽  
Vol 09 (04) ◽  
pp. 1850025 ◽  
Author(s):  
Jairo Rodríguez-Padilla ◽  
Daniel Olmos-Liceaga
Keyword(s):  
Wave Propagation ◽  
Numerical Methods ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Pseudospectral Method ◽  
Finite Difference Schemes ◽  
Cardiac Models ◽  
Computational Resources ◽  
Realistic Geometries

The implementation of numerical methods to solve and study equations for cardiac wave propagation in realistic geometries is very costly, in terms of computational resources. The aim of this work is to show the improvement that can be obtained with Chebyshev polynomials-based methods over the classical finite difference schemes to obtain numerical solutions of cardiac models. To this end, we present a Chebyshev multidomain (CMD) Pseudospectral method to solve a simple two variable cardiac models on three-dimensional anisotropic media and we show the usefulness of the method over the traditional finite differences scheme widely used in the literature.

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T121-T131 ◽  
Author(s):  
Tieyuan Zhu ◽  
Tong Bai
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Seismic Attenuation ◽  
Fourier Pseudospectral Method ◽  
Frequency Independent ◽  
Theoretical Predictions

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

scholarly journals A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection–Diffusion Problems with Boundary/Interior Layers

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1123
Author(s):  
Tianlong Ma ◽  
Lin Zhang ◽  
Fujun Cao ◽  
Yongbin Ge
Keyword(s):  
Present Method ◽  
Multigrid Method ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Finite Difference Schemes ◽  
Interior Layer ◽  
Convection Diffusion ◽  
Uniform Grids ◽  
Interior Layers ◽  
Diffusion Problems

Boundary or interior layer problems of high-dimensional convection–diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric problems. In this paper, we extended our previous work on the partial semi-coarsening multigrid method combined with the high-order compact (HOC) difference scheme for solving the two-dimensional (2D) convection–diffusion problems on non-uniform grids to the three-dimensional (3D) cases. The main merit of the present method is that the multigrid method on non-uniform grids can be performed with a different number of grids in different coordinate axes, which is more efficient than the multigrid method on non-uniform grids with the same number of grids in different coordinate axes. Numerical experiments are carried out to validate the accuracy and efficiency of the present method. It is shown that, without losing the high precision, the present method is very effective to reduce computing cost by cutting down the number of grids in the direction(s) which does/do not contain boundary or interior layer(s).

COUPLED 3D WAVE EQUATIONS WITH IRREGULAR FLUID-ELASTIC INTERFACE: THEORETICAL DEVELOPMENT

2002 ◽  
Vol 10 (04) ◽  
pp. 421-444 ◽  
Author(s):  
RAYMOND J. NAGEM ◽  
DING LEE
Keyword(s):  
Wave Propagation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Three Dimensional ◽  
Elastic Wave Propagation ◽  
Theoretical Development ◽  
Fluid Medium ◽  
Elastic Bottom ◽  
Elastic Interface ◽  
Irregular Interface

A coupled three-dimensional fluid-elastic wave propagation mathematical model has been developed to handle environmental interactions in the ocean between a fluid medium and an elastic bottom. The existing model combines three-dimensional fluid and elastic wave propagation models with the incorporation of a set of horizontal fluid-elastic interface conditions. This paper extends the above model to consider an irregular fluid-elastic interface. The theoretical development of the irregular fluid-elastic interface equations is presented. Comparison of the fluid-elastic irregular interface to the horizontal case is one of the major objectives of this paper. Another major objective of this paper is the construction of a complete set of three-dimensional fluid-elastic wave equations, including the irregular interface, in a form suitable for stable numerical solution.

Numerical studies of high-dimensional nonlinear viscous and nonviscous wave equations by using finite difference methods

2020 ◽  
Vol 11 (02) ◽  
pp. 2050008 ◽  
Author(s):  
Ding-Wen Deng ◽  
Zhu-An Wang
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Discrete Energy ◽  
Alternating Direction Implicit ◽  
Numerical Studies ◽  
Alternating Direction ◽  
Difference Methods

The numerical solutions of two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations via the unified alternating direction implicit (ADI) finite difference methods (FDMs) are obtained in this paper. By making use of the discrete energy method, it is proven that their numerical solutions converge to exact solutions with an order of two in both time and space with respect to [Formula: see text]-norm. Numerical results confirm that they are relatively accurate and high-resolution, and more successfully simulate the conservation of the energy for nonviscous equations, and the dissipation of the energy for viscous equation.

scholarly journals One-Way Wave Equation Derived from Impedance Theorem

Acoustics ◽  
2020 ◽  
Vol 2 (1) ◽  
pp. 164-171
Author(s):  
Oskar Bschorr ◽  
Hans-Joachim Raida
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Partial Differential Equations ◽  
Unique Solution ◽  
Wave Equations ◽  
Three Dimensional ◽  
Transverse Waves ◽  
Force Moment ◽  
Inherent Ambiguity ◽  
Dimensional Wave

The wave equations for longitudinal and transverse waves being used in seismic calculations are based on the classical force/moment balance. Mathematically, these equations are 2nd order partial differential equations (PDE) and contain two solutions with a forward and a backward propagating wave, therefore also called “Two-way wave equation”. In order to solve this inherent ambiguity many auxiliary equations were developed being summarized under “One-way wave equation”. In this article the impedance theorem is interpreted as a wave equation with a unique solution. This 1st order PDE is mathematically more convenient than the 2nd order PDE. Furthermore the 1st order wave equation being valid for three-dimensional wave propagation in an inhomogeneous continuum is derived.

A ROBUST HIGH-ORDER COMPACT METHOD FOR THE THREE DIMENSIONAL NONLINEAR BIHARMONIC EQUATIONS

2014 ◽  
Vol 11 (04) ◽  
pp. 1350065 ◽  
Author(s):  
SHUYING ZHAI ◽  
XINLONG FENG ◽  
YINNIAN HE
Keyword(s):  
Fourth Order ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
High Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Biharmonic Equations ◽  
Navier Stokes Equation ◽  
Unknown Variable

In this paper, a new family of fourth-order compact finite difference schemes are considered using coupled approach for numerical solutions of the three-dimensional (3D) linear biharmonic problems. A new fourth-order accurate algorithm is developed through the different composition of these schemes for 3D nonlinear biharmonic equations. And an optimal combination is found in numerical experiments. The main advantage of this algorithm is that it avoids the difficulties of constructing high order compact difference schemes for 3D nonlinear biharmonic equations. The numerical solutions of unknown variable and its first derivative and Laplacian are obtained. Finally, numerical experiments are conducted to show the solution accuracy and verify the validity of our new method, including the steady Navier–Stokes equation and Cahn–Hilliard equation.

scholarly journals On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations

2021 ◽  
Vol 2 (6) ◽  
Author(s):  
Hendrik Ranocha ◽  
Manuel Quezada de Luna ◽  
David I. Ketcheson
Keyword(s):  
Numerical Methods ◽  
Solitary Wave ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Broad Class ◽  
Dispersive Wave ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Nonlinear Dispersive Wave Equations ◽  
Nonlinear Dispersive Wave

AbstractWe study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.

A NEW PROCEDURE TO ACHIEVE REQUIRED ACCURACY IN COMPUTATIONAL OCEAN ACOUSTICS: THEORETICAL DEVELOPMENT

2012 ◽  
Vol 20 (04) ◽  
pp. 1250012 ◽  
Author(s):  
DING LEE ◽  
CHI-FANG CHEN
Keyword(s):  
Wave Propagation ◽  
Numerical Methods ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Theoretical Development ◽  
Acoustic Wave Propagation ◽  
Accuracy Requirement ◽  
Correct Procedure ◽  
Predictor Corrector ◽  
Ocean Acoustic Wave

Ocean acoustic wave propagation can be predicted by applying numerical methods to solve representative wave equations computationally. For this purpose, numerical methods have been introduced; a latest introduction was the Predictor-Corrector Method. An important question arises: Whether or not these numerical methods can produce satisfactory required accurate results? This may cause an accuracy concerned by the users. This paper introduces a new Predict-Correct Procedure to examine whether or not the result meets the accuracy requirement. If not, the procedure can improve the result until it becomes satisfactorily accurate. Discussions will be given on the mathematical and computational developments of the Predictor-Corrector Method as well as the Predict-Correct Procedure. Following that is a discussion on how the Predict-Correct Procedure works. An important part of this paper is devoted to show how this new procedure can achieve the goal of obtaining the required accurate prediction results.

scholarly journals Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 667
Author(s):  
Maryam Almutairi ◽  
Hamzeh Zureigat ◽  
Ahmad Izani Ismail ◽  
Ali Fareed Jameel
Keyword(s):  
Finite Difference ◽  
Fuzzy Number ◽  
Wave Equations ◽  
Fuzzy Numbers ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Finite Difference Schemes ◽  
Numerical Schemes ◽  
Implicit Schemes ◽  
Number Form

The use of fuzzy partial differential equations has become an important tool in which uncertainty or vagueness exists to model real-life problems. In this article, two numerical techniques based on finite difference schemes that are the centered time center space and implicit schemes to solve fuzzy wave equations were used. The core of the article is to formulate a new form of centered time center space and implicit schemes to obtain numerical solutions fuzzy wave equations in the double parametric fuzzy number approach. Convex normalized triangular fuzzy numbers are represented by fuzziness, based on a double parametric fuzzy number form. The properties of fuzzy set theory are used for the fuzzy analysis and formulation of the proposed numerical schemes followed by the new proof stability thermos under in the double parametric form of fuzzy numbers approach. The consistency and the convergence of the proposed scheme are discussed. Two test examples are carried out to illustrate the feasibility of the numerical schemes and the new results are displayed in the forms of tables and figures where the results show that the schemes have not only been effective for accuracy but also for reducing computational cost.

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