Numerical Differentiation by High Order Interpolation

1987 ◽  
Vol 8 (6) ◽  
pp. 979-987 ◽  
Author(s):  
Peter Hoffman ◽  
K. C. Reddy
Keyword(s):  
Numerical Differentiation ◽  
High Order ◽  
High Order Interpolation

scholarly journals On accurate high-order numerical derivatives computations for quantum chemistry purposes

2018 ◽  
Keyword(s):  
Quantum Chemistry ◽  
Finite Field ◽  
Numerical Differentiation ◽  
Field Method ◽  
High Order ◽  
Parameter Choice ◽  
Derivatives Of ◽  
Finite Field Method ◽  
Account Electron

Various molecular parameters in quantum chemistry could be computed as derivatives of energy over different arguments. Unfortunately, it is quite complicated to obtain analytical expression for characteristics that are of interest in the framework of methods that account electron correlation. Especially it relates to the coupled cluster (CC) theory. In such cases, numerical differentiation comes to rescue. This approach, like any other numerical method has empirical parameters and restrictions that require investigation. Current work is called to clarify the details of Finite-Field method usage for high-order derivatives calculation in CC approaches. General approach to the parameter choice and corresponding recommendations about numerical steadiness verification are proposed. As an example of Finite-Field approach implementation characterization of optical properties of fullerene passing process through the aperture of carbon nanotorus is given.

AUSM-Based High-Order Solution for Euler Equations

2012 ◽  
Vol 12 (4) ◽  
pp. 1096-1120 ◽  
Author(s):  
Angelo L. Scandaliato ◽  
Meng-Sing Liou
Keyword(s):  
Euler Equations ◽  
Splitting Method ◽  
High Order ◽  
Riemann Solvers ◽  
Carbuncle Phenomenon ◽  
Solution Accuracy ◽  
Matrix Vector ◽  
Approximate Riemann Solvers ◽  
Monotonicity Preserving ◽  
High Order Interpolation

AbstractIn this paper we demonstrate the accuracy and robustness of combining the advection upwind splitting method (AUSM), specifically AUSM+-UP, with high-order upwind-biased interpolation procedures, the weighted essentially non-oscillatory (WENO-JS) scheme and its variations, and the monotonicity preserving (MP) scheme, for solving the Euler equations. MP is found to be more effective than the three WENO variations studied. AUSM+-UP is also shown to be free of the so-called “carbuncle” phenomenon with the high-order interpolation. The characteristic variables are preferred for interpolation after comparing the results using primitive and conservative variables, even though they require additional matrix-vector operations. Results using the Roe flux with an entropy fix and the Lax-Friedrichs approximate Riemann solvers are also included for comparison. In addition, four reflective boundary condition implementations are compared for their effects on residual convergence and solution accuracy. Finally, a measure for quantifying the efficiency of obtaining high order solutions is proposed; the measure reveals that a maximum return is reached after which no improvement in accuracy is possible for a given grid size.

Assessment of Enhanced Multi-Dimensional Limiting Process for Multi-Dimensional Flows

2011 ◽  
Author(s):  
Kyu Hong Kim ◽  
Jung Ho Park
Keyword(s):  
Higher Order ◽  
High Order ◽  
Test Cases ◽  
Computational Domain ◽  
Interpolation Scheme ◽  
Local Extrema ◽  
Numerous Test ◽  
High Order Interpolation

In this paper, a new limiting process based on the Multi-dimensional Limiting Process, called enhanced Multi-dimensional Limiting Process is developed and tested with several cases. The enhanced Multi-dimensional Limiting Process, e-MLP has a number of useful features of MLP limiter such as multi-dimensional monotonicity and straightforward extensionality to higher order interpolation. It is applicable to local extrema and prevents excessive damping in a linear discontinuous region through application of appropriate limiting criteria. It is efficient because a limiting function is applied only to a discontinuous region. In addition, it is robust against shock instability due to the strict distinction of the computational domain and the use of regional information in a flux scheme as well as a high order interpolation scheme. The new limiting process was applied to numerous test cases. Through these tests, we could confirm that e-MLP enhances the accuracy and efficiency with both continuous and discontinuous multidimensional flows.

High-Order Interpolation Method for Overset Grid Based on Finite Volume Method

AIAA Journal ◽  
2011 ◽  
Vol 49 (7) ◽  
pp. 1387-1398 ◽  
Author(s):  
Kyung Rock Lee ◽  
Jung Ho Park ◽  
Kyu Hong Kim
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Interpolation Method ◽  
High Order ◽  
Overset Grid ◽  
Volume Method ◽  
High Order Interpolation ◽  
Grid Based

scholarly journals Inverse differential quadrature method: mathematical formulation and error analysis

2021 ◽  
Vol 477 (2248) ◽  
Author(s):  
Saheed O. Ojo ◽  
Luan C. Trinh ◽  
Hasan M. Khalid ◽  
Paul M. Weaver
Keyword(s):  
Differential Equations ◽  
Numerical Stability ◽  
Differential Quadrature Method ◽  
Mathematical Formulation ◽  
High Sensitivity ◽  
Numerical Differentiation ◽  
High Order ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Engineering Systems

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions.

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