Third-order methods for first-order hyperbolic partial differential equations

2003 ◽  
Vol 20 (1) ◽  
pp. 31-41
Author(s):  
T. A. Cheema ◽  
M. S. A. Taj ◽  
E. H. Twizell
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hyperbolic Partial Differential Equations ◽  
Third Order ◽  
Partial Differential ◽  
First Order

Eigenstructure of First-Order Velocity-Stress Equations for Waves in Elastic Solids of Trigonal 32 Symmetry

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Lixiang Yang ◽  
Yung-Yu Chen ◽  
S.-T. John Yu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Three Dimensional ◽  
Coefficient Matrix ◽  
Hyperbolic Partial Differential Equations ◽  
Wave Polarization ◽  
Partial Differential ◽  
Plane Wave Solution ◽  
Wave Speeds ◽  
First Order

This paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three 9×9 coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

Evaluation of Coupled Rheological Equations

2013 ◽  
Vol 433-435 ◽  
pp. 1943-1946
Author(s):  
Lei Hou ◽  
Pan Sun ◽  
Jun Jie Zhao ◽  
Lin Qiu ◽  
Han Lin Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Free Surface Flow ◽  
Internal Boundary Layer ◽  
Surface Flow ◽  
Hyperbolic Partial Differential Equations ◽  
Internal Boundary ◽  
Partial Differential ◽  
First Order ◽  
Super Convergence

Apart from the convergence of the first-order hyperbolic partial differential equations in rheological flow,this paper estimate the general behavior of the solution. By analyzing the coupled partial differential equations on a macroscopic scale the solution of free surface flow has been obtained. Its asymptotic estimate of the solution and super convergence are proposed in the internal boundary layer.

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