Numerical solutions of equations of cardiac wave propagation based on Chebyshev multidomain pseudospectral methods

2018 ◽  
Vol 151 ◽  
pp. 29-53
Author(s):  
Jairo Rodríguez-Padilla ◽  
Daniel Olmos-Liceaga
Keyword(s):  
Wave Propagation ◽  
Numerical Solutions ◽  
Pseudospectral Methods ◽  
Solutions Of Equations

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.

Numerical Solutions of Transients in Pneumatic Networks—Transmission-Line Calculations

1968 ◽  
Vol 35 (3) ◽  
pp. 588-595 ◽  
Author(s):  
S. Tsao
Keyword(s):  
Wave Propagation ◽  
Transmission Lines ◽  
Numerical Solutions ◽  
Navier Stokes ◽  
Damping Factor ◽  
Temperature Profiles ◽  
Hypothetical Case ◽  
One Dimensional ◽  
Simplifying Assumptions ◽  
Energy Equations

Equations governing the damped wave propagation along transmission lines are obtained from the Navier-Stokes and energy equations by making certain simplifying assumptions. The flow considered is essentially one-dimensional. However, radial variations of the velocity and temperature profiles must be considered, because the damping factor is directly dependent on them. The equations are integrated by numerical methods. A hypothetical case is computed as an example.

Elastic-wave propagation in two evenly-welded quarter-spaces

1971 ◽  
Vol 61 (5) ◽  
pp. 1119-1152
Author(s):  
Mario Ottaviani
Keyword(s):  
Wave Propagation ◽  
Elastic Wave ◽  
Numerical Solutions ◽  
Elastic Wave Propagation ◽  
Seismic Modeling ◽  
Interface Waves ◽  
Surface And Interface ◽  
Modeling Techniques ◽  
Vertical Displacements ◽  
Quarter Space

abstract This paper deals with elastic-wave propagation in two evenly-welded quarter-spaces. A compressional line source can be located at any point within either medium. The numerical solutions to this problem have been obtained by using the finite difference method. A computer program has been written to obtain synthetic seismograms of the horizontal and vertical displacements at all nodes of the superimposed grid, for the following cases: (a) elastic-wave propagation in a quarter-space, and (b) elastic-wave propagation in two quarter-spaces. Reflected, converted, transmitted, and diffracted phases are identified and interpreted. Surface and interface waves, originated at the corner by diffraction of the source pulse, are investigated as a function of the rigidity contrast and the velocity contrast between the two media and of the position of the source. Two-dimensional seismic modeling techniques have been used to provide a qualitative experimental verification of the numerical results.

Numerical solutions of equations describing nonisothermal flows of a real gas in tubes

1967 ◽  
Vol 13 (4) ◽  
pp. 290-293
Author(s):  
B. L. Krivoshein ◽  
E. M. Minskii ◽  
V. P. Radchenko ◽  
I. E. Khodanovich ◽  
M. G. Khublaryan
Keyword(s):  
Numerical Solutions ◽  
Real Gas ◽  
Nonisothermal Flows ◽  
Solutions Of Equations

Numerical Solutions of Polynomial Equations for the Eight-Position Synthesis of the Cylindrical-Spherical Dyad

2012 ◽  
Author(s):  
Chintien Huang ◽  
Chenning Hung ◽  
Kuenming Tien
Keyword(s):  
Rigid Body ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Quadratic Equation ◽  
Geometric Constraints ◽  
Numerical Continuation ◽  
Polynomial Equations ◽  
Quartic Polynomial ◽  
Solutions Of Equations ◽  
Position Synthesis

This paper investigates the numerical solutions of equations for the eight-position rigid-body guidance of the cylindrical-spherical (C-S) dyad. We seek to determine the number of finite solutions by using the numerical continuation method. We derive the design equations using the geometric constraints of the C-S dyad and obtain seven quartic polynomial equations and one quadratic equation. We then solve the system of equations by using the software package Bertini. After examining various specifications, including those with random complex numbers, we conclude that there are 804 finite solutions of the C-S dyad for guiding a body through eight prescribed positions. When designing spatial dyads for rigid-body guidance, the C-S dyad is one of the four dyads that result in systems of equal numbers of equations and unknowns if the maximum number of allowable positions is specified. The numbers of finite solutions in the syntheses of the other three dyads have been obtained previously, and this paper provides the computational kinematic result of the last unsolved problem, the eight-position synthesis of the C-S dyad.

A MATHEMATICAL MODEL FOR ANALYSIS OF WAVE PROPAGATION IN A LINEARIZED VERTICALLY NONUNIFORM PARTIALLY IONIZED GAS

1966 ◽  
Vol 44 (5) ◽  
pp. 1047-1065 ◽  
Author(s):  
Harold R. Raemer
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Mhd Waves ◽  
Mathematical Treatment ◽  
Algebraic Differential Equation ◽  
Neutral Gas ◽  
Partially Ionized ◽  
Analytical Approaches ◽  
Adiabatic State

Wave propagation in fully and partially ionized gases, with and without magnetic fields, has been treated by several workers; e.g., Tanenbaum and Mintzer (1962) obtained dispersion relations for a linearized and spatially uniform gas of electrons, positive ions, and neutrals. The present paper discusses the basic formulation and mathematical treatment of wave propagation in a linearized electron – ion – neutral gas, with static magnetic field, in which ambient-gas parameters vary arbitrarily vertically and are uniform horizontally.A standard formulation of the general problem is discussed via Boltzmann and Maxwell equations. By momentum-space averaging, the Boltzmann equation yields motion, continuity, and dynamic adiabatic state equations. These are combined to yield neutral and plasma equations of motion, continuity, and adiabatic state and a generalized Ohm's Law. Steady-state plane-wave solutions of the form exp[−i(ωt – kxx)] are assumed, reducing the x, y, and t dependence to algebraic relations, but the equations remain differential in z. The system consists of 10 simultaneous coupled ordinary first-order complex differential equations and 11 simultaneous complex algebraic equations in 21 complex unknowns.The second part of the paper is a discussion of the solution of this coupled algebraic differential equation system, equivalent to the system arising in the analysis of coupled linear electrical networks. Referring to the literature of differential equations and modern automatic control systems, various purely analytical approaches are discussed with emphasis on their deficiencies in obtaining practical numerical results with an arbitrary z variation. The Runge–Kutta step-by-step-procedure was invoked eventually and a Fortran program based on this technique was written. The program can be used to obtain accurate numerical solutions to many problems involving wave propagation in a linearized, vertically nonuniform electron – ion – neutral gas without requiring drastic simplifying assumptions for the vertical nonuniformity. This program can be used, by changing input parameter values, to treat such diverse problems as the perturbing effect of acoustic–gravity waves on ionospheric electron density, electromagnetic wave propagation in the vertically inhomogeneous ionosphere, MHD waves high in the ionosphere, or various kinds of wave propagation in prepared plasmas with a one-dimensional inhomogeneity. Numerical solutions for the acoustic – gravity wave – plasma interaction problem and their interpretation will be reported in a later paper.

scholarly journals A stable implementation measure of multi-transmitting formula in the numerical simulation of wave motion

PLoS ONE ◽  
2020 ◽  
Vol 15 (12) ◽  
pp. e0243979
Author(s):  
Jie Su ◽  
Zhenghua Zhou ◽  
Yuandong Li ◽  
Bing Hao ◽  
Qing Dong ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Numerical Solutions ◽  
Scattering Problem ◽  
Spherical Wave ◽  
Initial Boundary ◽  
Hyperbolic Partial Differential Equations ◽  
Initial Boundary Value Problems ◽  
Wave Source ◽  
Initial Boundary Value ◽  
Drift Instability

The Multi-Transmitting Formula (MTF) proposed by Liao et al. is a local artificial boundary condition widely used in numerical simulations of wave propagation in an infinite medium, while the drift instability is usually caused in its numerical implementation. In view of a physical interpretation of the Gustafsson, Kreiss and Sundström criterion on numerical solutions of initial-boundary value problems in the hyperbolic partial differential equations, the mechanism of the drift instability of MTF was discussed, and a simple measure for eliminating the drift instability was proposed by introducing a modified operator into the MTF. Based on the theory of spherical wave propagation and damping effect of medium, the physical implication on modified operator was interpreted. And the effect of the modified operator on the reflection coefficient of MTF was discussed. Finally, the validity of the proposed stable implementation measure was verified by numerical tests of wave source problem and scattering problem.

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