In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In additional, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.
In \cite{christlieb2019kernel}, the authors developed a class of high-order numerical schemes for the Hamilton-Jacobi (H-J) equations, which are unconditionally stable, yet take the form of an explicit scheme. This paper extends such schemes, so that they are more effective at capturing sharp gradients, especially on nonuniform meshes. In particular, we modify the weighted essentially non-oscillatory (WENO) methodology in the previously developed schemes by incorporating an exponential basis and adapting the previously developed nonlinear filters used to control oscillations. The main advantag…
The ideal Magnetohydrodynamics (MHD) equations are challenging because one needs to maintain the divergence free condition, $\nabla \cdot \Bv = 0$. Many numerical methods have been developed to enforce this condition. In this work, we further our work on mesh aligned constrained transport by developing a new kernel based approach for the vector potential in 2D and 3D. The approach for solving the vector potential is based on the method of lines transpose and is A-stable, eliminating the need for diffusion limiters needed in our previous work in 3D. The work presented here is an improvement ove…
We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a parabolic type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in cer…
An unconditionally stable algorithm for generalised thermoelasticity based on operator-splitting and time-discontinuous Galerkin finite element methods
University of Cape Town（开普敦大学）；Centre for Research in Computational and Applied Mechanics (CERECAM)（计算与应用力学研究中心）
In this paper, three efficient ensemble algorithms are proposed for fast-solving the random fluid-fluid interaction model. Such a model can be simplified as coupling two heat equations with random diffusion coefficients and a friction parameter due to its complexity and uncertainty. We utilize the Monte Carlo method for the coupled model with random inputs to derive some deterministic fluid-fluid numerical models and use the ensemble idea to realize the fast computation of multiple problems. Our remarkable feature of these algorithms is employing the same coefficient matrix for multiple linear…
An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn-Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., \Ass{1}: $0<r_k:=τ_k/τ_{k-1}< r_{\max}\approx 4.8645$, we establish a rigorous error estimate in $H^1$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that t…
Maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions and nonlinear operator preserves for all time a uniform pointwise bound in absolute value. It has been a challenging problem on how to design unconditionally MBP-preserving high-order accurate time-stepping schemes for these equations. In this paper, we combine the integrating factor Runge-Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (…
We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be appl…
In this paper, a free boundary problem modelling the growth of tumor is considered. The model includes two reaction-diffusion equations modelling the diffusion of nutrient and drug in the tumor and three hyperbolic equations describing the evolution of three types of cells (i.e. proliferative cells, quiescent cells and dead cells) considered in the tumor. Due to the fact that in the real situation, the subdiffusion of nutrient and drug in the tumor can be found, we have changed the reaction-diffusion equations to the fractional ones to consider other conditions and study a more general and rel…
The ubiquity of semilinear parabolic equations has been illustrated in their numerous applications ranging from physics, biology, to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t=\mathcal{L}u+f[u]$ with $\mathcal{L}$ being a linear dissipative operator and $f$ being a nonlinear operator in space, namely a time-invariant maximum bound principle, in the sense that the time-dependent solution $u$ preserves for all time a uniform pointwise bound in absolute value imposed by its init…
We consider machine-learning of time-dependent quantities of interest derived from solution trajectories of parabolic partial differential equations. For large-scale or long-time integration scenarios, where using a full order model (FOM) to generate sufficient training data is computationally prohibitive, we propose an adaptive hierarchy of intermediate Reduced Basis reduced order models (ROM) to augment the FOM training data by certified ROM training data required to fit a kernel model.
A posteriori error estimates in the maximum norm for parabolic problems
University of Kentucky（肯塔基大学）；University of Sussex（萨塞克斯大学）；University of Crete（克里特大学）；Institute for Applied and Computational Mathematics, Foundation for Research and Technology-Hellas（希腊研究与技术基金会应用与计算数学研究所）
Recent theoretical and experimental advances show that the inertia of magnetization emerges at sub-picoseconds and contributes to the ultrafast magnetization dynamics which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial Landau-Lifshitz-Gilbert (iLLG) equation is proposed to model the ultrafast magnetization dynamics. Mathematically, the LLG equation is a nonlinear system of parabolic type with (possible) degeneracy. However, the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicat…
