Numerical solution of wave equation using higher order methods

Nečasová, Gabriela; Šátek, Václav; Kunovský, Jiří

doi:10.1063/1.5043964

Cited by 1 publication

(4 citation statements)

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“…In this paper, the Lax-Wendroff difference scheme with Richardson extrapolation numerical method is developed to solve a one-dimensional wave equation with integral boundary conditions given in Eq. (1). Hyperbolic equations such as the wave equation in Eq.…”

Section: Methodsmentioning

confidence: 99%

“…Consider that the following linear onedimensional wave equation with integral boundary condition considered in [2,5,10,16] given by: 𝑢𝑢 �� 𝑢𝑢 �� 𝑥𝑥𝑥 𝑥𝑥�, �𝑥𝑥𝑥 𝑥𝑥� ∈ �𝑎𝑎𝑥 𝑎𝑎� � �0𝑥 𝑇𝑇� (1) with initial and boundary condition respectively…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…( 6) into Eq. ( 9) we obtain, (10) where 𝛼𝛼 � � �� , 𝐺𝐺 � � � 2𝛼𝛼 � 𝑞𝑞�𝑥𝑥 � 𝑎 𝑎𝑎 � � and 𝜏𝜏 � � � 𝛼𝛼𝜏𝜏 � � 2𝛼𝛼 � 𝜏𝜏 � � 𝜏𝜏 � is local truncation error term of obtained difference equation for the full discretization of the one-dimensional hyperbolic partial differential equation given in (1). This local truncation error is:…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…The presence of an airwave can be suppressed by setting the sound speed in the air equal to that at the surface, using constant extrapolation [9]. Again the numerical methods have been widely implemented to model acoustic and hydrodynamic waves [1]. This method of solving for the elastic wave eigenmodes in acoustic waveguides of an arbitrary cross-section is presented.…”

Section: Introductionmentioning

confidence: 99%

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Lax-Wendroff Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected To Integral Condition

Koroche

2021

ijmph

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In this paper, the Lax-Wendroff difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion and the Lax-Wendroff difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L � and L � norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

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Section: Methodsmentioning

confidence: 99%