The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of the elastic properties of r...
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The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of the elastic properties of rocks and the dissipative effects due to internal friction and viscosity;acoustic waves propagating through biological tissues, where both elastic and viscous effects play a significant role. We propose a stable and high-order finitedifference method for solving the governing equations. By designing the spatial discretization with the summation-by property, we prove stability by deriving a discrete energy estimate. In addition, we derive error estimates for problems with constant coefficients using the normal mode analysis and problems with variable coefficients using the energy method. Numerical examples are presented to demonstrate the stability and accuracy properties of the developed method.
This study explores the potential of utilizing hardware built for Machine Learning (ML) tasks as a platform for solving linear Partial Differential Equations via numerical methods. We examine the feasibility, benefits...
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This study explores the potential of utilizing hardware built for Machine Learning (ML) tasks as a platform for solving linear Partial Differential Equations via numerical methods. We examine the feasibility, benefits, and obstacles associated with this approach. Given an Initial Boundary Value Problem (IBVP) and a finitedifference method, we directly compute stencil coefficients and assign them to the kernel of a convolution layer, a common component used in ML. The convolution layer's output can be applied iteratively in a stencil loop to construct the solution of the IBVP. We describe this stencil loop as a TensorFlow (TF) program and use a Google Cloud instance to verify that it can target ML hardware and to profile its behavior and performance. We show that such a solver can be implemented in TF, creating opportunities in exploiting the computational power of ML accelerators for numerics and simulations. Furthermore, we discover that the primary issues in such implementations are under-utilization of the hardware and its low arithmetic precision. We further identify data movement and boundary condition handling as potential future bottlenecks, underscoring the need for improvements in the TF backend to optimize such computational patterns. Addressing these challenges could pave the way for broader applications of ML hardware in numerical computing and simulations.
The purpose of this study is to determine what finite-difference algorithms are best used in numerical simulation of two-dimensional single-phase saturated porous media flows when the models have a nondiagonal symmetr...
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The purpose of this study is to determine what finite-difference algorithms are best used in numerical simulation of two-dimensional single-phase saturated porous media flows when the models have a nondiagonal symmetric tenser for the mobility (or hydraulic conductivity) that has nontrivial jump discontinuities along lines that are not aligned with the coordinate axes. Such problems arise naturally in many modeling situations and, in addition, when simpler problems are studied using adaptive grids. The answer is surprising, the simplest finite-difference method, called the MAC Scheme with Linear Averaging, performs nearly as well as most other algorithms over a wide range of problems. A new algorithm, called the Full Harmonic Averaged Scheme, is significantly more costly to use, but does perform better than the simplest scheme in certain interesting cases. The simplest finite-difference method is compared to some finite-element simulations taken from the literature;the finite-difference algorithm performs better. Many of the conclusions of the paper rest on testing the algorithms on a new class of problems with analytic solutions. The problems have a nondiagonal mobility tenser and can have a jump discontinuity of arbitrary height.
In this report, finite difference methods of orders 2 and 4 are developed and analysed for the solution of a two-point boundary value problem associated with a fourth- order linear differential equation. A sufficient ...
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In this report, finite difference methods of orders 2 and 4 are developed and analysed for the solution of a two-point boundary value problem associated with a fourth- order linear differential equation. A sufficient condition guaranteeing a unique solution of the boundary value problem is also given. Numerical results for a typical boundary value problem are tabulated and compared with the shooting technique using the fourth-order Runge-Kutta method.
Several finite difference methods are proposed for the infinitesimal generator of 1D asymmetric alpha-stable Levy motions, based on the fact that the operator becomes a multiplier in the spectral space. These methods ...
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Several finite difference methods are proposed for the infinitesimal generator of 1D asymmetric alpha-stable Levy motions, based on the fact that the operator becomes a multiplier in the spectral space. These methods take the general form of a discrete convolution, and the coefficients (or the weights) in the convolution are chosen to approximate the exact multiplier after appropriate transform. The accuracy and the associated advantages/disadvantages are also discussed, providing some guidance on the choice of the right scheme for practical problems, like in the calculation of mean exit time for random processes governed by general asymmetric alpha-stable motions.
Adaptive methods for solving systems of partial differential equations have become widespread. Much of the effort has focused on finite element methods. In this paper modified finitedifference approximations are obta...
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Adaptive methods for solving systems of partial differential equations have become widespread. Much of the effort has focused on finite element methods. In this paper modified finitedifference approximations are obtained for grids with irregular nodes. The modifications are required to ensure consistency and stability. Asymptotically exact a posteriori error estimates of the spatial error are presented for the finitedifference method. These estimates are derived from interpolation estimates and are computed using central difference approximations of second derivatives of the solution at grid nodes. The interpolation error estimates are shown to converge for irregular grids while the a posteriori error estimates are shown to converge for uniform grids. Computational results demonstrate the convergence of the finitedifference method and a posteriori error estimates for cases not covered by the theory.
In this paper, we derive results about the numerical performance of multi-point (moving average) finitedifference formulas for the differentiation of non-exact data. In particular, we show that multi-point differenti...
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In this paper, we derive results about the numerical performance of multi-point (moving average) finitedifference formulas for the differentiation of non-exact data. In particular, we show that multi-point differentiators can be constructed which are asymptotically unbiased and have a bounded amplification factor as the steplength decreases and the number of points increases.
Basic properties of some finitedifference schemes for two-dimensional nonlinear dispersive equations for hydrodynamics of surface waves are considered. It is shown that stability conditions for difference schemes of ...
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Basic properties of some finitedifference schemes for two-dimensional nonlinear dispersive equations for hydrodynamics of surface waves are considered. It is shown that stability conditions for difference schemes of shallow water equations are qualitatively different in the cases the dispersion is taken into account, or not. The difference in the behavior of phase errors in one-and two-dimensional cases is pointed out. Special attention is paid to the numerical algorithm based on the splitting of the original system of equations into a nonlinear hyperbolic system and a scalar linear equation of elliptic type.
For solving the regime switching utility maximization, Fu et al. (Eur J Oper Res 233:184-192, 2014) derive a framework that reduce the coupled Hamilton-Jacobi-Bellman (HJB) equations into a sequence of decoupled HJB e...
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For solving the regime switching utility maximization, Fu et al. (Eur J Oper Res 233:184-192, 2014) derive a framework that reduce the coupled Hamilton-Jacobi-Bellman (HJB) equations into a sequence of decoupled HJB equations through introducing a functional operator. The aim of this paper is to develop the iterative finite difference methods (FDMs) with iteration policy to the sequence of decoupled HJB equations derived by Fu et al. (2014). The convergence of the approach is proved and in the proof a number of difficulties are overcome, which are caused by the errors from the iterative FDMs and the policy iterations. Numerical comparisons are made to show that it takes less time to solve the sequence of decoupled HJB equations than the coupled ones.
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