arXiv subjects

Takahito Kashiwabara

Publications and source records attributed to Takahito Kashiwabara.

At least 19 recordsLinked to original sources

arXiv2026

Convergence Analysis of the Upwind Difference Methods for Hamilton-Jacobi-Bellman Equations

This paper investigates the convergence properties of the upwind difference scheme for the Hamilton--Jacobi--Bellman (HJB) equation, a central partial differential equation in optimal control theory. First, assuming the existence of a classical solution, we show that the numerical solution converges to the true solution with a first-order rate with respect to the time step. This result complements the square-root rate established in previous studies for viscosity solutions. Second, by exploiting the correspondence between HJB equations and conservation laws, we prove the convergence of the optimal control input. This analysis is crucial for practical applications where the control input is the primary quantity of interest, yet it has rarely been addressed in previous studies. Finally, we confirm the validity of our theoretical results through numerical experiments on typical control problems.

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arXiv2026

Single exponential $H^1$-upper bounds for the primitive equations

The three dimensional primitive equations with full viscosity are considered in a horizontally periodic box $Ω$, which are subject to either the homogeneous Neumann or Dirichlet conditions on the upper and bottom parts of the boundary. For a strong solution $v$ with initial data $a$, we establish \emph{a priori} bounds in $L^\infty(0, \infty; H^1(Ω)) \cap L^2(0, \infty; \dot H^2(Ω))$, the exponential part of which is $\exp(C \|a\|_{L^2(Ω)}^2)$. This is in contrast to the upper bounds reported in the existing literature that are double exponential. Furthermore, the uniform-in-time estimate for the Neumann condition case, in which the Poincaré inequality is unavailable for $v$, seems to be new.

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arXiv2026

Local-in-time strong solvability of Navier--Stokes type variational inequalities by Rothe's method

We consider parabolic variational inequalities in a Hilbert space $V$, which have a non-monotone nonlinearity of Navier--Stokes type represented by a bilinear operator $B: V \times V \to V'$ and a monotone type nonlinearity described by a convex, proper, and lower-semicontinuous functional $φ: V \to (-\infty, +\infty]$. Existence and uniqueness of a local-in-time strong solution in a maximal-$L^2$-regularity class and in a Kiselev--Ladyzhenskaya class are proved by discretization in time (also known as Rothe's method), provided that a corresponding stationary Stokes problem admits a regularity structure better than $V$ (which is typically $H^2$-regularity in case of the Navier--Stokes equations). Since we do not assume the cancelation property $\left< B(u, v), v \right> = 0$, in applications we may allow for broader boundary conditions than those treated by the existing literature.

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arXiv2025

$H^2$-regularity for stationary and non-stationary Bingham problems with perfect slip boundary condition

$H^2$-spatial regularity of stationary and non-stationary problems for Bingham fluids formulated with the pseudo-stress tensor is discussed. The problem is mathematically described by an elliptic or parabolic variational inequality of the second kind, to which weak solvability in the Sobolev space $H^1$ is well known. However, higher regularity up to the boundary in a bounded smooth domain seems to remain open. This paper indeed shows such $H^2$-regularity if the problems are supplemented with the so-called perfect slip boundary condition and if the yield stress vanishes on the boundary. For the stationary Bingham--Stokes problem, the key of the proof lies in a priori estimates for a regularized problem avoiding investigation of higher pressure regularity, which seems difficult to get in the presence of a singular diffusion term. The $H^2$-regularity for the stationary case is then directly applied to establish strong solvability of the non-stationary Bingham--Navier--Stokes problem, based on discretization in time and on the truncation of the nonlinear convection term.

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arXiv2025

Optimal $L^\infty$-error estimate for isoparametric finite element method in a smooth domain

We consider the isoparametric finite element method (FEM) for the Poisson equation in a smooth domain with the homogeneous Dirichlet boundary condition. Because the boundary is curved, standard triangulated meshes do not exactly fit it. Thereby we need to introduce curved elements if better accuracy than linear FEM is desired, which necessitates the use of isoparametric FEMs. We establish optimal rate of convergence $O(h^{k+1})$ in the $L^\infty$-norm for $k \ge 2$, by extending the approach of our previous work [Kashiwabara and Kemmochi, Numer.\ Math.\ \textbf{144}, 553--584 (2020)] developed for Neumann boundary conditions and $k = 1$.

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arXiv2025

The three limits of the hydrostatic approximation

The primitive equations are derived from the $3D$-Navier-Stokes equations by the hydrostatic approximation. Formally, assuming an $\varepsilon$-thin domain and anisotropic viscosities with vertical viscosity $ν_z=\mathcal{O}(\varepsilon^γ)$ where $γ=2$, one obtains the primitive equations with full viscosity as $\varepsilon\to 0$. Here, we take two more limit equations into consideration: For $γ<2$ the $2D$-Navier-Stokes equations are obtained. For $γ>2$ the primitive equations with only horizontal viscosity $-Δ_H$ as $\varepsilon\to 0$. Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally $ν_z=\varepsilon^2 δ$ and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for $\varepsilon,δ\to 0$. The flexibility of our methods is also illustrated by the convergence for $δ\to \infty$ and $\varepsilon\to 0$ to the $2D$-Navier-Stokes equations.

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arXiv2025

On periodic traveling wave solutions with or without phase transition to the Navier-Stokes-Korteweg and the Euler-Korteweg equations

The Navier-Stokes-Korteweg and the Euler-Korteweg equations are considered in isothermal setting. These are diffuse interface models of two-phase flow. For the Navier-Stokes-Korteweg equations, we show that there is no periodic traveling wave solution with phase transition although there exists a non-constant periodic traveling wave solution with no phase transition. For the Euler-Korteweg equations, we show that there always exists a periodic traveling wave solution with phase transition for any period if the Korteweg relaxation parameter is small compared with the period provided that the available energy is double-well type. We also show that such a periodic traveling wave solution tends to a monotone traveling wave solution as the period tends to infinity under suitable spatial translation. Our numerical experiment suggests that there is periodic traveling wave with phase transition which is stable under periodic perturbation for small viscosity but it seems that this is a transition pattern.

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arXiv2024

Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations

Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by $ h $ and $ Δt $ respectively, we prove an error estimate $ O(Δt^3 + \frac{h^4}{Δt}) $ in $ L^2 $ norm theoretically, which justifies the above-mentioned prediction if $ h = O(Δt) $. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the $ L^2 $ projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation.

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arXiv2024

The well-posedness and regularity of the Non-stationary Stokes and Navier-Stokes equations with the friction-type interface condition

The friction-type interface condition (FIC) is introduced to describe the phenomenon of the slip and leak of fluid flow on the interface happens only when the difference of stress force is above a threshold. The FIC involves the subdifferential and can be regarded as an intermediate form of the Dirichlet and the Neumann boundary conditions. This work is devoted to the well-posedness of the non-stationary (Navier-)Stokes equations with FIC in 2D and 3D, the weak forms of which are parabolic variational inequalities of the second type. We establish the existence theorems using the regularization technique and the Galerkin method. For the Stokes case, we prove the global unique existence and investigate the $H^2$ regularity. In the case of 2D Navier-Stokes equation, we show the global unique existence of the weak and strong solutions, respectively. For the Navier-Stokes case in 3D, we demonstrate the global existence of the weak solution and the local unique existence of the strong solution.

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arXiv2024

Discrete maximal regularity for the discontinuous Galerkin time-stepping method without logarithmic factor

Maximal regularity is a kind of a priori estimates for parabolic-type equations and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green's function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.

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arXiv2024

An Uncertainty-aware, Mesh-free Numerical Method for Kolmogorov PDEs

This study introduces an uncertainty-aware, mesh-free numerical method for solving Kolmogorov PDEs. In the proposed method, we use Gaussian process regression (GPR) to smoothly interpolate pointwise solutions that are obtained by Monte Carlo methods based on the Feynman-Kac formula. The proposed method has two main advantages: 1. uncertainty assessment, which is facilitated by the probabilistic nature of GPR, and 2. mesh-free computation, which allows efficient handling of high-dimensional PDEs. The quality of the solution is improved by adjusting the kernel function and incorporating noise information from the Monte Carlo samples into the GPR noise model. The performance of the method is rigorously analyzed based on a theoretical lower bound on the posterior variance, which serves as a measure of the error between the numerical and true solutions. Extensive tests on three representative PDEs demonstrate the high accuracy and robustness of the method compared to existing methods.

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arXiv2023

Finite element analysis of a generalized Robin boundary value problem in curved domains based on the extension method

A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $Ω$ is a general smooth domain with a curved boundary, we need to introduce an approximate domain $Ω_h$ and to address issues owing to the domain perturbation $Ω\neq Ω_h$. In contrast to the transformation approach used in existing studies, we employ the extension approach, which is easier to handle in practical computation, in order to construct a numerical scheme. Assuming that approximate domains and function spaces are given by isoparametric finite elements of order $k$, we prove the optimal rate of convergence in the $H^1$- and $L^2$-norms. A numerical example is given for the piecewise linear case $k = 1$.

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arXiv2023

A fictitious-play finite-difference method for linearly solvable mean field games

An iterative finite difference scheme for mean field games (MFGs) is proposed. The target MFGs are derived from control problems for multidimensional systems with advection terms. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.

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arXiv2022

Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type, and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model.

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arXiv2021

Justification of the Hydrostatic Approximation of the Primitive Equations in Anisotropic Space $L^\infty_H L^q_{x_3}(\Torus^3)$

The primitive equations are fundamental models in geophysical fluid dynamics and derived from the scaled Navier-Stokes equations. In the primitive equations, the evolution equation to the vertical velocity is replaced by the so-called hydrostatic approximation. In this paper, we give a justification of the hydrostatic approximation by the scaled Navier-Stoke equations in anisotropic spaces $L^\infty_H L^q_{x_3} (\Torus^3)$ for $q \geq 1$.

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arXiv2021

Model Predictive Mean Field Games for Controlling Multi-Agent Systems

When controlling multi-agent systems, the trade-off between performance and scalability is a major challenge. Here, we address this difficulty by using mean field games (MFGs), which is a framework that deduces the macroscopic dynamics describing the density profile of agents from their microscopic dynamics. To effectively use the MFG, we propose a model predictive MFG (MP-MFG), which estimates the agent population density profile with using kernel density estimation and manages the input generation with model predictive control. The proposed MP-MFG generates control inputs by monitoring the agent population at each time step, and thus achieves higher robustness than the conventional MFG. Numerical results show that the MP-MFG outperforms the MFG when the agent model has modeling errors or the number of agents in the system is small.

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arXiv2021

A robust discontinuous Galerkin scheme on anisotropic meshes

Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

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arXiv2020

The Hydrostatic Approximation for the Primitive Equations by the Scaled Navier-Stokes Equations under the No-Slip Boundary Condition

In this paper we justify the hydrostatic approximation of the primitive equations in the maximal $L^p$-$L^q$-setting in the three-dimensional layer domain $Ω= \Torus^2 \times (-1, 1)$ under the no-slip (Dirichlet) boundary condition in any time interval $(0, T)$ for $T>0$. We show that the solution to the scaled Navier-Stokes equations with Besov initial data $u_0 \in B^{s}_{q,p}(Ω)$ for $s > 2 - 2/p + 1/ q$ converges to the solution to the primitive equations with the same initial data in $\mathbb{E}_1 (T) = W^{1, p}(0, T ; L^q (Ω)) \cap L^p(0, T ; W^{2, q} (Ω)) $ with order $O(ε)$ where $(p,q) \in (1,\infty)^2$ satisfies $ \frac{1}{p} \leq \min \bracket{ 1 - 1/q, 3/2 - 2/q }$. The global well-posedness of the scaled Navier-Stokes equations in $\mathbb{E}_1 (T)$ is also proved for sufficiently small $ε>0$. Note that $T = \infty$ is included.

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