Chunmei Su

Weizhu Bao, Remi Carles, Chunmei Su et al.

We present a regularized finite difference method for the logarithmic Schrödinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. $\ln ρ\to -\infty$ when $ρ\rightarrow 0^+$ with $ρ=|u|^2$ being the density and $u$ being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schrödinger equation (RLogSE) is proposed with a small regularization parameter $0<\varepsilon\ll 1$ and linear convergence is established between the solutions of RLogSE and LogSE in term of $\varepsilon$. Then a semi-implicit finite difference method is presented for discretizing the RLogSE and error estimates are established in terms of the mesh size $h$ and time step $τ$ as well as the small regularization parameter $\varepsilon$. Finally numerical results are reported to confirm our error bounds.

Weizhu Bao, Rémi Carles, Chunmei Su et al.

We present and analyze two numerical methods for the logarithmic Schr{ö}dinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank-Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schr{ö}dinger equation (RLogSE) with a small regularized parameter 0 < $ε$ $\ll$ 1 is adopted to approximate the LogSE with linear convergence rate O($ε$). Then we use the Lie-Trotter splitting integrator to solve the RLogSE and establish its error bound O($τ$ 1/2 ln($ε$ --1)) with $τ$ > 0 the time step, which implies an error bound at O($ε$ + $τ$ 1/2 ln($ε$ --1)) for the LogSE by the Lie-Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE.

Weizhu Bao, Chunmei Su

We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter $\varepsilon \in (0,1]$, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. $0<\varepsilon \ll 1$, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with $O(\varepsilon)$-wavelength in time and $O(1)$-wavelength in space as well as outgoing initial layers in space at speed $O(1/\varepsilon)$. This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ. By adapting an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at $O(h^2+τ^2/\varepsilon)$ and $O(h^2+τ+\varepsilon)$ with $h$ mesh size and $τ$ time step. Thus we obtain a uniform error bound at $O(h^2+τ)$ for $0<\varepsilon\le 1$. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and $\varepsilon$-dependent error bounds between the solutions of KGZ and its limiting model when $\varepsilon\to0^+$. Finally, numerical results are reported to confirm our error bounds.

Weizhu Bao, Chunmei Su

We present a uniformly accurate finite difference method and establish rigorously its uniform error bounds for the Zakharov system (ZS) with a dimensionless parameter $0<\varepsilon\le 1$, which is inversely proportional to the speed of sound. In the subsonic limit regime, i.e., $0<\varepsilon\ll 1$, the solution propagates highly oscillatory waves and/or rapid outgoing initial layers due to the perturbation of the wave operator in ZS and/or the incompatibility of the initial data which is characterized by two nonnegative parameters $α$ and $β$. Specifically, the solution propagates waves with $O(\varepsilon)$- and $O(1)$-wavelength in time and space, respectively, and amplitude at $O(\varepsilon^{\min\{2,α,1+β\}})$ and $O(\varepsilon^α)$ for well-prepared ($α\ge1$) and ill-prepared ($0\le α<1$) initial data, respectively. This high oscillation of the solution in time brings significant difficulties in designing numerical methods and establishing their error bounds, especially in the subsonic limit regime. A uniformly accurate finite difference method is proposed by reformulating ZS into an asymptotic consistent formulation and adopting an integral approximation of the oscillatory term. By adapting the energy method and using the limiting equation via a nonlinear Schrödinger equation with an oscillatory potential, we rigorously establish two independent error bounds and obtain error bounds at $O(h^2+τ^{4/3})$ and $O(h^2+τ^{1+\fracα{2+α}})$ for well-prepared and ill-prepared initial data, respectively, which are uniform in both space and time for $0<\varepsilon\le 1$ and optimal at the second order in space. Numerical results are reported to demonstrate that our error bounds are sharp.

Alexander Ostermann, Chunmei Su

We introduce two exponential-type integrators for the "good" Bousinessq equation. They are of orders one and two, respectively, and they require lower regularity of the solution compared to the classical exponential integrators. More precisely, we will prove first-order convergence in Hrfor solutions in H^{r+1} with r > 1/2 for the derived first-order scheme. For the second integrator, we prove second-order convergence in Hrfor solutions in H^{r+3} with r > 1/2 and convergence in L2for solutions in H^3. Numerical results are reported to illustrate the established error estimates. The experiments clearly demonstrate that the new exponential-type integrators are favorable over classical exponential integrators for initial data with low regularity.

Alexander Ostermann, Chunmei Su

We propose an explicit numerical method for the periodic Korteweg-de Vries equation. Our method is based on a Lawson-type exponential integrator for time integration and the Rusanov scheme for Burgers' nonlinearity. We prove first-order convergence in both space and time under a mild Courant-Friedrichs-Lewy condition $τ=O(h)$, where $τ$ and $h$ represent the time step and mesh size, respectively. Numerical examples illustrating our convergence result are given.

Chunmei Su, Zhiping Li

The approximation properties of a quadratic iso-parametric finite element method for a typical cavitation problem in nonlinear elasticity are analyzed. More precisely, (1) the finite element interpolation errors are established in terms of the mesh parameters; (2) a mesh distribution strategy based on an error equi-distribution principle is given; (3) the convergence of finite element cavity solutions is proved. Numerical experiments show that, in fact, the optimal convergence rate can be achieved by the numerical cavity solutions.

Xiuhui Guo, Wei Jiang, Chunmei Su

We propose a family of high-order local discontinuous Galerkin (LDG) methods, built on a parametric representation and coupled with a semi-implicit backward Euler time discretization, for isotropic and anisotropic curve-shortening flows. The spatial LDG formulation introduces auxiliary variables and carefully designed numerical fluxes which inherit the underlying variational structure. We prove the unconditional energy dissipation for the semi-discrete scheme, and establish the well-posedness for the fully discrete scheme under mild assumptions. For $P^k$ approximations, the LDG method achieves high-order spatial convergence; extensive numerical experiments confirm optimal $(k+1)$-order accuracy when the surface energy is isotropic or weakly anisotropic. Compared to classical parametric finite element methods (PFEM), the proposed LDG schemes do not need to rely on good mesh distributions or auxiliary symmetrized surface energy matrices for strongly anisotropic surface energy cases, and remain numerically stable on severely degraded meshes that typically cause PFEMs failure. This intrinsic stability enables effective capture of complex geometric evolution and sharp corner singularities produced by strong anisotropy. The approach thus provides a flexible and reliable framework for the numerical simulation of a broader class of geometric flows.

Chunmei Su, Xiaofei Zhao

We present a uniformly and optimally accurate numerical method for solving the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters $0<ε\le1$ and $0<γ\le 1$, which are inversely proportional to the plasma frequency and the acoustic speed, respectively. In the simultaneous high-plasma-frequency and subsonic limit regime, i.e. $ε<γ\to 0^+$, the KGZ system collapses to a cubic Schrödinger equation, and the solution propagates waves with $O(ε^2)$-wavelength in time and meanwhile contains rapid outgoing initial layers with speed $O(1/γ)$ in space due to the incompatibility of the initial data. By presenting a multiscale decomposition of the KGZ system, we propose a multiscale time integrator Fourier pseduospectral method which is explicit, efficient and uniformly accurate for solving the KGZ system for all $0<ε<γ\leq1$. Numerical results are reported to show the efficiency and accuracy of scheme. Finally, the method is applied to investigate the convergence rates of the KGZ system to its limiting models when $ε<γ\to 0^+$.

Chunmei Su, Zhiping Li

The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient $\det \nabla u$ is required to be positive, is a natural physical constraint in elasticity as well as in many other fields. It is well known that the constraint can often cause serious difficulties in both theoretical analysis and numerical computation, especially when the material is subject to large deformations. In this paper, we derive a set of sufficient and necessary conditions for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations. The result provides a practical quantitative guide for meshing in the neighborhood of a cavity and shows that the orientation-preservation can be achieved with a reasonable number of total degrees of freedom by the quadratic iso-parametric finite element method.