R. Martin¹, D. Komatitsch^1,2, S. D. Gedney³, E. Bruthiaux^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.1, pp. 17-42, 2010, DOI:10.3970/cmes.2010.056.017

Abstract Unsplit convolutional perfectly matched layers (CPML) for the velocity and stress formulation of the seismic wave equation are classically computed based on a second-order finite-difference time scheme. However it is often of interest to increase the order of the time-stepping scheme in order to increase the accuracy of the algorithm. This is important for instance in the case of very long simulations. We study how to define and implement a new unsplit non-convolutional PML called the Auxiliary Differential Equation PML (ADE-PML), based on a high-order Runge-Kutta time-stepping scheme and optimized at grazing incidence. We demonstrate More >

Hrvoje Gotovac¹, Vedrana Kozulić², Blaž Gotovac¹

CMC-Computers, Materials & Continua, Vol.15, No.3, pp. 173-198, 2010, DOI:10.3970/cmc.2010.015.173

Abstract The space-time Adaptive Fup Collocation Method (AFCM) for solving boundary-initial value problems is presented. To solve the one-dimensional initial boundary value problem, we convert the problem into a two-dimensional boundary value problem. This quasi-boundary value problem is then solved simultaneously in the space-time domain with a collocation technique and by using atomic Fup basis functions. The proposed method is a generally meshless methodology because it requires only the addition of collocation points and basis functions over the domain, instead of the classical domain discretization and numerical integration. The grid is adapted progressively by setting the More >

Hossein M. Shodja^{1, 2, 3}, Alireza Hashemian^{2, 4}

The International Conference on Computational & Experimental Engineering and Sciences, Vol.13, No.3, pp. 57-58, 2009, DOI:10.3970/icces.2009.013.057

Abstract Gradient reproducing kernel particle method (GRKPM) is a meshless technique which incorporates the first gradients of the function into the reproducing equation of RKPM. Therefore, in two-dimensional space GRKPM introduces three types of shape functions rather than one. The robustness of GRKPM's shape functions is established by reconstruction of a third-order polynomial. To enforce the essential boundary conditions (EBCs), GRKPM's shape functions are modified by transformation technique. By utilizing the modified shape functions, the weak form of the nonlinear evolutionary Buckley-Leverett (BL) equation is discretized in space, rendering a system of nonlinear ordinary differential equations More >

Chein-Shan Liu¹, Weichung Yeih², Chung-Lun Kuo³, Satya N. Atluri⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.1, pp. 47-72, 2009, DOI:10.3970/cmes.2009.053.047

Abstract Iterative algorithms for solving a system of nonlinear algebraic equations (NAEs): F_i(x_j) = 0, i, j = 1,... ,n date back to the seminal work of Issac Newton. Nowadays a Newton-like algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation. However, this type of algorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrix ∂F_i/∂x_j and its inverse at each iterative step. In addition, the Newton-like methods restrict one to construct an iteration procedure for n-variables… More >

Dan Gordon¹, Rachel Gordon²

CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.1, pp. 23-46, 2009, DOI:10.3970/cmes.2009.053.023

Abstract Linear systems with very large off-diagonal elements and discontinuous coefficients (LODC systems) arise in some modeling cases, such as those involving heterogeneous media. Such problems are usually solved by domain decomposition methods, but these can be difficult to implement on unstructured grids or when the boundaries between subdomains have a complicated geometry. Gordon and Gordon have shown that Björck and Elfving's (sequential) CGMN algorithm and their own block-parallel CARP-CG are very robust and efficient on strongly convection dominated cases (but without discontinuous coefficients). They have also shown that scaling the equations by dividing each equation… More >

Hua Li¹, Shantanu S. Mulay¹, Simon See²

CMES-Computer Modeling in Engineering & Sciences, Vol.48, No.1, pp. 43-82, 2009, DOI:10.3970/cmes.2009.048.043

Abstract Differential Quadrature (DQ) is one of the efficient derivative approximation techniques but it requires a regular domain with all the points distributed only along straight lines. This severely restricts the DQ while solving the irregular domain problems discretized by the random field nodes. This limitation of the DQ method is overcome in a proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. The RDQ method extends the applicability of the DQ technique over the irregular or regular domains discretized using the random field nodes by approximating a function value with the fixed… More >

Hung-Chang Lee¹, Chein-Shan Liu²

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.1, pp. 1-26, 2009, DOI:10.3970/cmes.2009.041.001

Abstract The group-preserving schemes developed by Liu (2001) for integrating ordinary differential equations system were adopted the Cayley transform and Padé approximants to formulate the Lie group from its Lie algebra. However, the accuracy of those schemes is not better than second-order. In order to increase the accuracy by employing the group-preserving schemes on ordinary differential equations, according to an efficient technique developed by Runge and Kutta to raise the order of accuracy from the Euler method, we combine the Runge-Kutta method on the group-preserving schemes to obtain the higher-order numerical methods of group-preserving type. They More >

Chunhai Kou¹, Ye Yan², Jian Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.39, No.3, pp. 301-318, 2009, DOI:10.3970/cmes.2009.039.301

Abstract In the paper, stability for fractional order differential equations is studied. Then the result obtained is applied to analyse the stability of equilibrium for the model of HIV. More >

R. Prosser¹

FDMP-Fluid Dynamics & Materials Processing, Vol.5, No.4, pp. 411-424, 2009, DOI:10.3970/fdmp.2009.005.411

Abstract A novel wavelet-based method for the simulation of reacting flows on adaptive meshes is presented. The method is based on a subtraction algorithm, wherein the wavelet coefficients are calculated from the low resolution up (as opposed to the standard top-down approach). The advantage of this new method is that it allows the calculation of wavelet coefficients on sparse grids, and thus lends itself more readily to adaptive computational meshes than does the traditional wavelet algorithm. The approach is used to simulate a one-dimensional laminar pre-mixed flame with different Lewis numbers. The computational grid is adapted More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.10, No.1, pp. 97-116, 2009, DOI:10.3970/cmc.2009.010.097

Abstract A new numerical method is proposed for solving the delay ordinary differential equations (DODEs) under multiple time-varying delays or state-dependent delays. The finite difference scheme is used to approximate the ODEs, which together with the initial conditions constitute a system of nonlinear algebraic equations (NAEs). Then, a Fictitious Time Integration Method (FTIM) is used to solve these NAEs. Numerical examples confirm that the present approach is highly accurate and efficient with a fast convergence. More >