Convergence Analysis of a Second Order Convex Splitting Scheme for the Modified Phase Field Crystal Equation
نویسندگان
چکیده
In this paper we provide a detailed convergence analysis for an unconditionally energy stable, second order accurate convex splitting scheme for the modified phase field crystal equation, a generalized damped wave equation for which the usual phase field crystal equation is a special degenerate case. The fully discrete, fully second order finite difference scheme in question was derived in a recent work [A. Baskaran et al., J. Comput. Phys., 250 (2013), pp. 270–292]. An introduction of a new variable ψ, corresponding to the temporal derivative of the phase variable φ, could bring an accuracy reduction in the formal consistency estimate, because of the hyperbolic nature of the equation. A higher order consistency analysis by an asymptotic expansion is performed to overcome this difficulty. In turn, second order convergence in both time and space is established in a discrete L∞(0, T ;H3) norm.
منابع مشابه
An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation
We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the grad...
متن کاملEnergy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation
In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first orde...
متن کاملAn Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation
We present an unconditionally energy stable finite-difference scheme for the phase field crystal equation. The method is based on a convex splitting of a discrete energy and is semiimplicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step size. We present local-in-time error est...
متن کاملGlobal Smooth Solutions of the Three-dimensional Modified Phase Field Crystal Equation
Abstract. The Modified Phase Field Crystal (MPFC) equation, a generalized damped wave equation for which the usual Phase Field Crystal (PFC) equation is a special case, is analyzed in detail in three dimensions. A time-discrete numerical scheme, based on a convex splitting for the functional energy, is utilized to construct an approximate solution, which is then shown to converge to a solution ...
متن کاملThree-Phase Modeling of Dynamic Kill in Gas-Condensate Well Using Advection Upstream Splitting Method Hybrid Scheme
Understanding and modeling of three-phase transient flow in gas-condensate wells play a vital role in designing and optimizing dynamic kill procedure of each well that needs to capture the discontinuities in density, geometry, and velocity of phases but also the effect of temperature on such parameters. In this study, two-phase Advection-Upstream-Splitting-Method (AUSMV) hybrid scheme is extend...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM J. Numerical Analysis
دوره 51 شماره
صفحات -
تاریخ انتشار 2013