Search

Parabolic initial–boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution. For those problems, specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution. This article investigates characteristics of the θ -method and introduces a variable- θ method as a synergistic combination of the Crank–Nicolson (CN) method ( θ = 1 ∕ 2 ) and the implicit method ( θ = 1 ). It suppresses spurious oscillations, by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data. An effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations (the wobble set); the resulting variable- θ method is analyzed for its accuracy and stability. Various numerical examples are given to verify its effectiveness.

This article introduces recursive Heaviside step functions, as a potential of the known universe, for the first time in the history of mathematics, science, and engineering. In modern cosmology, various bouncing models have been suggested based on the postulation that the current universe is the result of the collapse of a previous universe. However, all Big Bounce models leave unanswered the question of what powered inflation. Recursive Heaviside step functions are analyzed to represent the warpage of spacetime during the crunch-bounce transition. In particular, the time shift appeared during the transition is modeled in the form of recursive Heaviside step functions and suggested as a possible answer for the immeasurable energy appeared for the Big Bounce.

This article introduces a new fourth-order implicit time-stepping scheme for the numerical solution of the acoustic wave equation, as a variant of the conventional modified equation method. For an efficient simulation, the scheme incorporates a locally one-dimensional (LOD) procedure having the splitting error of O(@Dt^4). Its stability and accuracy are compared with those of the standard explicit fourth-order scheme. It has been observed from various experiments for 2D problems that (a) the computational cost of the implicit LOD algorithm is only about 40% higher than that of the explicit method, for the problems of the same size, (b) the implicit LOD method produces less dispersive solutions in heterogeneous media, and (c) its numerical stability and accuracy match well those of the explicit method.

This article is concerned with accurate and efficient numerical methods for solving viscous and nonviscous wave equations. A three-level second-order implicit algorithm is considered without introducing auxiliary variables. As a perturbation of the algorithm, a locally one-dimensional (LOD) procedure which has a splitting error not larger than the truncation error is suggested to solve problems of diagonal diffusion tensors in cubic domains efficiently. Both the three-level algorithm and its LOD procedure are proved to be unconditionally stable. An error analysis is provided for the numerical solution of viscous waves. Numerical results are presented to show the accuracy and efficiency of the new algorithms for the propagation of acoustic waves and of microscale heat transfer.

It is often noted that the Helmholtz equation is extremely difficult to solve, in particular, for high-frequency solutions for heterogeneous media. Since stability for second-order discretization methods requires one to choose at least 10–12 grid points per wavelength, the discrete problem on the possible coarsest mesh is huge. In a realistic simulation, one is required to choose 20–30 points per wavelength to achieve a reasonable accuracy; this problem is hard to solve. This article is concerned with the high-frequency asymptotic decomposition of the wavefield for an efficient and accurate simulation for the high-frequency numerical solution of the Helmholtz equation. It has been numerically verified that the new method is accurate enough even when one chooses 4–5 grid points per wavelength.

This article is concerned with the fourth-order compact scheme for an accurate simulation of acoustic waves in the frequency domain. The compact schemes have been known for a long time; however, they have exhibited difficulties, in particular, in dealing with general boundary conditions. This article introduces a new formulation for the compact scheme for the Helmholtz equation and an effective strategy of incorporating absorbing boundary conditions. It has been numerically verified that the resulting compact scheme is fourth order in general heterogeneous media and improves the accuracy of the numerical solution dramatically, by more than two digits, over the standard second-order scheme.

Seismic techniques incorporating high frequency asymptotic representation of the 3D elastic Green’s function require efficient solution methods for the computation of traveltimes. For finite difference eikonal solvers, upwind differences are requisite to sharply resolve discontinuities in the traveltime derivatives. In anisotropic media, the direction of energy propagation is not in general tangent to the wavefront normal, while finite difference eikonal solvers compute the solution based on the traveltime gradients and wavefront normal. Local convexity of the wavefronts in transverse isotropic (TI) media is proved to show that wavefront normal determines the upwind direction of the energy propagation. The eikonal equations for the traveltimes in TI media of a generally inclined symmetry axis (ITI) are derived in a way that the eikonal solvers fit conveniently. A stable, second-order, shock-capturing, upwind finite difference scheme is suggested for solving ITI eikonal equations in regular grids in 3D. Numerical experiments are presented to demonstrate the efficiency of the algorithm.

A frequency-stepping algorithm for solving multifrequency (acoustic) wave propagation is considered. A two-grid method is employed for the problems of single frequency. For high frequency applications, the coarse grid problem is still huge, since one has to choose at least six to eight grid points per wavelength for a stability reason. The coarse grid problem is solved by a nonoverlapping domain decomposition (DD) method. The solution of the former frequency problem is used as the initial guess for the solution of the next larger frequency problem. Such an algorithm turns out to be efficient for multifrequency, as well as single-frequency problems, as shown in numerical results. Also, it is easily parallelizable with a high efficiency.

In this paper, we consider parallel iterative algorithms for solving the Helmholtz wave equation employing nonoverlapping domain decomposition techniques. A modified Robin interface condition incorporated with an iteration parameter is used to communicate the data near the interfaces. An automatic and non-expensive strategy for finding efficient iteration parameters is discussed in detail. Numerical results carried out on an nCUBE2 are given to demonstrate the effectiveness of the method.

A parallelizable iterative procedure based on domain decomposition techniques is defined and analyzed for finite difference approximate solutions for the Helmholtz problem. An automatic efficient strategy for choosing the algorithm parameter is presented. Numerical results are reported.