Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.9, No.3, pp. 229-252, 2009, DOI:10.3970/cmc.2009.009.229

Abstract When the given input data are corrupted by an intensive noise, most numerical methods may fail to produce acceptable numerical solutions. Here, we propose a new numerical scheme for solving the Burgers equation forward in time and backward in time. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ )u(x,t) =: v(x,t,τ), such that the original Burgers equation is written as a new parabolic type partial differential equation in the space of (x,t,τ). A fictitious damping coefficient can be used to strengthen the stability in the numerical integration of a semi-discretized… More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.8, No.2, pp. 53-66, 2008, DOI:10.3970/cmc.2008.008.053

Abstract Proposed is a time-marching algorithm to solve a nonlinear system of complementarity equations: P_i(x_j) ≥ 0, Q_i(x_j) ≥ 0 , P_i(x_j)Q_i(x_j) = 0, i, j = 1,...,n, resulting from a discretization of nonlinear obstacle problem. We transform the above nonlinear complementarity problem (NCP) into a nonlinear algebraic equations (NAEs) system: F_i(x_j) = 0 with the aid of the Fischer-Burmeister NCP-function. Such NAEs are semi-smooth, highly nonlinear and usually implicit, being hard to handle by the Newton-like method. Instead of, a first-order system of ODEs is derived through a fictitious time equation. The time-stepping equations are obtained by applying a numerical… More >

Chih-Wen Chang¹

CMC-Computers, Materials & Continua, Vol.21, No.2, pp. 87-106, 2011, DOI:10.3970/cmc.2011.021.087

Abstract We address a new numerical approach to deal with these multi-dimensional backward wave problems (BWPs) in this study. A fictitious time τ is utilized to transform the dependent variable u(x, y, z, t) into a new one by (1+τ)u(x, y, z, t)=: v(x, y, z, t, τ), such that the original wave equation is written as a new hyperbolic type partial differential equation in the space of (x, y, z, t, τ). Besides, a fictitious viscous damping coefficient can be employed to strengthen the stability of numerical integration of the discretized equations by using a group preserving scheme. Several numerical… More >

Chih-Wen Chang¹

CMC-Computers, Materials & Continua, Vol.19, No.3, pp. 285-314, 2010, DOI:10.3970/cmc.2010.019.285

Abstract In this article, we propose a new numerical approach for solving these multi-dimensional nonlinear and nonhomogeneous backward heat conduction problems (BHCPs). A fictitious time t is employed to transform the dependent variable u(x, y, z, t) into a new one by (1+t)u(x, y, z, t)=: v(x, y, z, t, t), such that the original nonlinear and nonhomogeneous heat conduction equation is written as a new parabolic type partial differential equation in the space of (x, y, z, t, t). In addition, a fictitious viscous damping coefficient can be used to strengthen the stability of numerical integration of the discretized equations… More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.15, No.3, pp. 221-240, 2010, DOI:10.3970/cmc.2010.015.221

Abstract We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: D_t^αu + εuu_x = vu_xx + ηD_x^βu, 0 < α, β ≤ 1, and u_t + D_*^β(D_*^1-βu)²/2 = vu_xx, 0 < β ≤ 1. The time-fractional derivative D_t^αu and space-fractional derivative D_x^βu are defined in the Caputo sense, while D_*^βu is the Riemann-Liouville space-fractional derivative. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ)^γu(x,t) =: v(x,t,τ), where 0 < γ ≤ 1 is a parameter, such that the original equation is written as a new functional-differential… 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 threshold as a direct measure… More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.11, No.1, pp. 15-32, 2009, DOI:10.3970/cmc.2009.011.015

Abstract Motivated by the evolutionary and dissipative properties of parabolic type partial differential equation (PDE), Liu (2008a) has proposed a natural and mathematically equivalent approach by transforming the quasilinear elliptic PDE into a parabolic one. However, the above paper only considered a rectangular domain in the plane, and did not treat the difficulty arisen from the quasilinear PDE defined in an arbitrary plane domain. In this paper we propose a new technique of internal and boundary residuals in a fictitious rectangular domain, which are driving forces for the ordinary differential equations based on the Fictitious Time Integration Method (FTIM). Several numerical… 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 >