Takayasu Matsuo

Stig Larsson, Takayasu Matsuo, Klas Modin et al.

The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational Derivative Method (DVDM) on a rectangular domain discretized with a regular, structured, orthogonal grid. We present numerical experiments to support our claims: we investigate the preservation of energy and linear momenta, the reversibility, and the empirical convergence of the schemes. The quality of our schemes is finally tested by simulating the interaction of singular wave fronts.

Yuto Miyatake, David Cohen, Daisuke Furihata et al.

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.

Daisuke Furihata, Shun Sato, Takayasu Matsuo

We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of difference operators is essential to the discrete conservation law. Unfortunately, however, when we employ the standard central difference operator, the simplest one, the numerical solutions often suffer from undesirable spatial oscillations. In this letter, we propose a novel "average-difference method," which is tougher against such oscillations, and combine it with an existing conservative method. Theoretical and numerical analysis in the linear case show the superiority of the proposed method.

Shun Sato, Takayasu Matsuo

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative.

Kaito Sato, Shun Sato, Takayasu Matsuo

Multi-symplectic diamond schemes proposed by McLachlan and Wilkins (2015) provide a framework for the numerical integration of Hamiltonian partial differential equations, combining local implicitness with high-order accuracy and discrete multi-symplectic conservation laws. Despite these advantages, their behavior beyond a limited class of model equations remains poorly understood, and numerical difficulties may arise depending on the underlying multi-symplectic formulation. In this paper, we present a systematic stability analysis framework for diamond schemes applied to general multi-symplectic PDEs. The approach consists of three stages. First, we identify structural inconsistency of the local diamond update using Dulmage--Mendelsohn decomposition, revealing cases in which the scheme is intrinsically unsolvable. Second, we introduce a graph-based error-propagation analysis that yields a necessary stability condition by detecting negative cycles in a weighted directed graph. Third, for equations that pass the preliminary tests, we derive eigenvalue-based timestep restrictions providing sufficient conditions for stability. The analysis leads to a comprehensive classification of multi-symplectic PDEs according to whether diamond schemes are structurally inconsistent, unconditionally unstable, or conditionally stable. In particular, we show that benchmark equations such as the Korteweg--de Vries equation are intrinsically incompatible with the diamond update, while systems including the nonlinear Dirac and ``good'' Boussinesq equations admit stability regimes under mild timestep scaling. Extensive numerical experiments confirm the theoretical predictions and demonstrate the practical implications of the proposed framework. Our results clarify fundamental limitations of diamond schemes and provide practical guidelines for their reliable application to new PDE models.

Kimiaki Kinugasa, Yuto Miyatake, Takayasu Matsuo

The numerical integration of the Benjamin and Benjamin--Ono equations are considered. They are non-local partial differential equations involving the Hilbert transform, and due to this, so far quite few structure-preserving integrators have been proposed. In this paper, a new reformulation of the equations is stated, and new structure-preserving discretizations are proposed based on it. Numerical experiments confirm the effectiveness of the proposed integrators.