Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

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Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽

10.3390/math9131483 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1483

Author(s):

Shanqin Chen

Keyword(s):

Large Time ◽

Hyperbolic Equations ◽

Strong Stability ◽

Matrix Exponential ◽

Linear Component ◽

Nonlinear Hyperbolic Equations ◽

Time Step ◽

Integrating Factor ◽

The Matrix ◽

Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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Distributions of processes governed by hyperbolic equations when also boundary conditions are assumed

Probability Theory and Mathematical Statistics ◽

10.1515/9783112319321-035 ◽

1994 ◽

pp. 605-612

Keyword(s):

Boundary Conditions ◽

Hyperbolic Equations

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On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

Georgian Mathematical Journal ◽

10.1515/gmj.2008.555 ◽

2008 ◽

Vol 15 (3) ◽

pp. 555-569

Author(s):

Tariel Kiguradze

Keyword(s):

Hyperbolic Equation ◽

Boundary Value Problems ◽

Fourth Order ◽

Hyperbolic Equations ◽

Boundary Value ◽

Sufficient Conditions ◽

Nonlinear Hyperbolic Equation ◽

Well Posedness ◽

Nonlinear Hyperbolic Equations ◽

Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

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