Unified Framework for Evaluating Numerical Integration Methods Through Characteristic Root Distortion in Linear Ordinary Differential Equations Article Swipe
Takuya Yamakawa
,
Yosuke Kurihara
,
Kazuyuki Kobayashi
,
Kajiro Watanabe
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.1109/access.2025.3598377
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.1109/access.2025.3598377
Analytically solving complex or large-scale differential equations is often difficult or even impossible, making numerical integration methods indispensable. However, as all numerical integration methods approximate a continuous-time equation using a discrete-time model, they inevitably introduce errors into the numerical solution. This study presents a unified framework for evaluating the numerical errors introduced by various integration methods. The integration formulae are expressed as equations involving differential and difference operators applied to the solution of the equation. This study applies integration methods to constant-coefficient linear ordinary differential equations (ODEs) and adopt an approach in which these expressions are transformed using Laplace and z-transforms, deriving characteristic equations that capture the essential properties of widely used integration methods. Based on the characteristic equations, distortions of the characteristic root —real and conjugate complex values that represent the fundamental properties of linear ODEs due to integration methods have been evaluated, and regions where stable roots become numerically unstable have also been identified. Additionally, transfer functions were estimated from the numerical solutions, and the distorted characteristic roots were corrected using the inverse of the approximation formula. The proximity of the restored roots to the original roots served as a quantitative indicator of the accuracy and reliability of each integration method. This unified evaluation approach revealed the distinct characteristics of widely used integration methods and offered a generalizable basis for interpreting errors in numerical solutions.
Related Topics
Concepts
Distortion (music)
Root (linguistics)
Exponential integrator
Ordinary differential equation
Mathematics
Applied mathematics
Differential algebraic equation
Computer science
Mathematical analysis
Differential equation
Computer network
Amplifier
Linguistics
Philosophy
Bandwidth (computing)
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1109/access.2025.3598377
- OA Status
- gold
- References
- 22
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- 10
- OpenAlex ID
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All OpenAlex metadata
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https://doi.org/10.1109/access.2025.3598377Digital Object Identifier
- Title
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Unified Framework for Evaluating Numerical Integration Methods Through Characteristic Root Distortion in Linear Ordinary Differential EquationsWork title
- Type
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2025Year of publication
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2025-01-01Full publication date if available
- Authors
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Takuya Yamakawa, Yosuke Kurihara, Kazuyuki Kobayashi, Kajiro WatanabeList of authors in order
- Landing page
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https://doi.org/10.1109/access.2025.3598377Publisher landing page
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YesWhether a free full text is available
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goldOpen access status per OpenAlex
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https://doi.org/10.1109/access.2025.3598377Direct OA link when available
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Distortion (music), Root (linguistics), Exponential integrator, Ordinary differential equation, Mathematics, Applied mathematics, Differential algebraic equation, Computer science, Mathematical analysis, Differential equation, Computer network, Amplifier, Linguistics, Philosophy, Bandwidth (computing)Top concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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| abstract_inverted_index.due | 137 |
| abstract_inverted_index.for | 46, 221 |
| abstract_inverted_index.the | 37, 48, 70, 73, 106, 116, 121, 131, 162, 166, 173, 176, 182, 186, 195, 208 |
| abstract_inverted_index.ODEs | 136 |
| abstract_inverted_index.This | 40, 75, 203 |
| abstract_inverted_index.also | 153 |
| abstract_inverted_index.been | 142, 154 |
| abstract_inverted_index.each | 200 |
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| abstract_inverted_index.they | 32 |
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| abstract_inverted_index.basis | 220 |
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| abstract_inverted_index.using | 28, 97, 172 |
| abstract_inverted_index.where | 146 |
| abstract_inverted_index.which | 92 |
| abstract_inverted_index.(ODEs) | 86 |
| abstract_inverted_index.become | 149 |
| abstract_inverted_index.errors | 35, 50, 223 |
| abstract_inverted_index.linear | 82, 135 |
| abstract_inverted_index.making | 13 |
| abstract_inverted_index.model, | 31 |
| abstract_inverted_index.served | 189 |
| abstract_inverted_index.stable | 147 |
| abstract_inverted_index.values | 128 |
| abstract_inverted_index.widely | 110, 212 |
| abstract_inverted_index.Laplace | 98 |
| abstract_inverted_index.applied | 68 |
| abstract_inverted_index.applies | 77 |
| abstract_inverted_index.capture | 105 |
| abstract_inverted_index.complex | 2, 127 |
| abstract_inverted_index.inverse | 174 |
| abstract_inverted_index.method. | 202 |
| abstract_inverted_index.methods | 16, 23, 79, 140, 215 |
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| abstract_inverted_index.regions | 145 |
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| abstract_inverted_index.various | 53 |
| abstract_inverted_index.However, | 18 |
| abstract_inverted_index.accuracy | 196 |
| abstract_inverted_index.approach | 90, 206 |
| abstract_inverted_index.deriving | 101 |
| abstract_inverted_index.distinct | 209 |
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| abstract_inverted_index.formula. | 178 |
| abstract_inverted_index.formulae | 58 |
| abstract_inverted_index.methods. | 55, 113 |
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| abstract_inverted_index.restored | 183 |
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| abstract_inverted_index.conjugate | 126 |
| abstract_inverted_index.corrected | 171 |
| abstract_inverted_index.difficult | 9 |
| abstract_inverted_index.distorted | 167 |
| abstract_inverted_index.equation. | 74 |
| abstract_inverted_index.equations | 6, 62, 85, 103 |
| abstract_inverted_index.essential | 107 |
| abstract_inverted_index.estimated | 160 |
| abstract_inverted_index.expressed | 60 |
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| abstract_inverted_index.functions | 158 |
| abstract_inverted_index.indicator | 193 |
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| abstract_inverted_index.involving | 63 |
| abstract_inverted_index.numerical | 14, 21, 38, 49, 163, 225 |
| abstract_inverted_index.operators | 67 |
| abstract_inverted_index.proximity | 180 |
| abstract_inverted_index.represent | 130 |
| abstract_inverted_index.solution. | 39 |
| abstract_inverted_index.difference | 66 |
| abstract_inverted_index.equations, | 118 |
| abstract_inverted_index.evaluated, | 143 |
| abstract_inverted_index.evaluating | 47 |
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| abstract_inverted_index.introduced | 51 |
| abstract_inverted_index.properties | 108, 133 |
| abstract_inverted_index.solutions, | 164 |
| abstract_inverted_index.solutions. | 226 |
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| abstract_inverted_index.distortions | 119 |
| abstract_inverted_index.expressions | 94 |
| abstract_inverted_index.fundamental | 132 |
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| abstract_inverted_index.large-scale | 4 |
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| abstract_inverted_index.interpreting | 222 |
| abstract_inverted_index.quantitative | 192 |
| abstract_inverted_index.Additionally, | 156 |
| abstract_inverted_index.approximation | 177 |
| abstract_inverted_index.discrete-time | 30 |
| abstract_inverted_index.generalizable | 219 |
| abstract_inverted_index.z-transforms, | 100 |
| abstract_inverted_index.characteristic | 102, 117, 122, 168 |
| abstract_inverted_index.indispensable. | 17 |
| abstract_inverted_index.characteristics | 210 |
| abstract_inverted_index.continuous-time | 26 |
| abstract_inverted_index.constant-coefficient | 81 |
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| countries_distinct_count | 1 |
| institutions_distinct_count | 4 |
| citation_normalized_percentile.value | 0.43783645 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |