Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives **...
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Transcript of Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives **...
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Finite Difference Methods
Definitions
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![Page 2: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/2.jpg)
Finite Difference MethodsApproximate derivatives
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** difference between exact derivative and its approximation is the truncation error
![Page 3: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/3.jpg)
Finite Difference MethodsGiven a smooth function (i.e., a function that is continuous and has continuous derivatives), the Taylor series approximates the value of a function at one point based on the value of the function and its derivatives at another, nearby point
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![Page 4: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/4.jpg)
Finite Difference MethodsFirst-order Taylor series:
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solving for the first derivative
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![Page 5: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/5.jpg)
Finite Difference Methods
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![Page 6: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/6.jpg)
Finite Difference Methods
Exercise: write the Taylor series expansion for q(x - x) and solve for the first differential...
![Page 7: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/7.jpg)
Finite Difference Methods
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![Page 8: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/8.jpg)
Finite Difference Methods
Backward time difference: explicit
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![Page 9: Finite Difference Methods Definitions. Finite Difference Methods Approximate derivatives ** difference between exact derivative and its approximation.](https://reader036.fdocuments.in/reader036/viewer/2022083008/56649f435503460f94c62c11/html5/thumbnails/9.jpg)