Finite-difference schemes for nonlinear wave equation that inherit energy conservation property

AbstractWe propose two general finite-difference schemes that inherit energy conservation property from nonlinear wave equations, such as the nonlinear Klein–Gordon equation (NLKGE). One of proposed schemes is implicit and another is explicit. Many studies exist on FDSs that inherit energy conservation property from NLKGE and we can derive all of their schemes from the proposed general schemes in this paper. The most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts, which implies that the inherited properties are satisfied exactly. Because of this the derived schemes are expected to be numerically stable and yield solutions converging to PDE solutions. We make new FDSs for Fermi–Pasta–Ulam equation, string vibration equation, Shimoji–Kawai equation (SKE) and Ebihara equation and verify numerically the inheritance of the energy conservation property for NLKGE and SKE

Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation

We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension $d\le 3$. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.Comment: 25 page

Structure-preserving method on Voronoi cells

When we want to use reference points located arbitrarily in two- or three-dimensional regions, it is essentially difficult to design some structure-preserving methods. The reason is that we should discretize some Gauss-Green formulae keeping some mathematical properties in that situations. Based on Voronoi-Delaunay triangulations, we can find some beautiful discrete Gauss-Green formulae and apply them to design some structure-preserving numerical methods. In the talk, we will indicate those formulae and their proofs in detail and the obtained discrete variational derivative methods based on Voronoi cells. Furthermore, if it is possible, we will show some relaxed structure-preserving methods to decrease computation cost.Non UBCUnreviewedAuthor affiliation: Osaka UniversityFacult

Discrete variational derivative method: a structure-preserving numerical method for partial differential equations

Nonlinear Partial Differential Equations (PDEs) have become increasingly important in the description of physical phenomena. Unlike Ordinary Differential Equations, PDEs can be used to effectively model multidimensional systems. The methods put forward in Discrete Variational Derivative Method concentrate on a new class of ""structure-preserving numerical equations"" which improves the qualitative behaviour of the PDE solutions and allows for stable computing. The authors have also taken care to present their methods in an accessible manner, which means that the book will be useful to enginee

Composing a surrogate observation operator for sequential data assimilation

In data assimilation, state estimation is not straightforward when the observation operator is unknown. This study proposes a method for composing a surrogate operator when the true operator is unknown. A neural network is used to improve the surrogate model iteratively to decrease the difference between the observations and the results of the surrogate model. A twin experiment suggests that the proposed method outperforms approaches that tentatively use a specific operator throughout the data assimilation process

Geometric numerical integrators for Hunter–Saxton-like equations

We present novel geometric numerical integrators for Hunter--Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter--Saxton equation, the modified Hunter--Saxton equation, and the two-component Hunter--Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators