Numerical Algorithms, ISSN 1017-1398, 11/2018, Volume 79, Issue 3, pp. 787 - 800

Two derivative Runge-Kutta methods are Runge-Kutta methods for problems of the form y ′ = f(y) that include the second derivative y ″ = g(y) = f ′(y)f(y) and...

Two derivative Runge-Kutta methods | Algorithms | Algebra | Numerical Analysis | Computer Science | Numeric Computing | Theory of Computation | Trigonometrical fitting | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATIONS | 2-STEP HYBRID METHODS | STABILITY | International trade | Information science | Analysis | Methods

Two derivative Runge-Kutta methods | Algorithms | Algebra | Numerical Analysis | Computer Science | Numeric Computing | Theory of Computation | Trigonometrical fitting | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATIONS | 2-STEP HYBRID METHODS | STABILITY | International trade | Information science | Analysis | Methods

Journal Article

Journal of Mathematical Chemistry, ISSN 0259-9791, 3/2018, Volume 56, Issue 3, pp. 799 - 812

A family of modified two-derivative Runge–Kutta (MTDRK) methods for the integration of the Schrödinger equation are obtained. Two new three-stage and fifth...

Theoretical and Computational Chemistry | Chemistry | Error analysis | Physical Chemistry | Two-derivative Runge–Kutta method | Phase-lag | Math. Applications in Chemistry | Schrödinger equation | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | Two-derivative Runge-Kutta method | CHEMISTRY, MULTIDISCIPLINARY | FAMILY | Error analysis (Mathematics) | Analysis | Methods

Theoretical and Computational Chemistry | Chemistry | Error analysis | Physical Chemistry | Two-derivative Runge–Kutta method | Phase-lag | Math. Applications in Chemistry | Schrödinger equation | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | Two-derivative Runge-Kutta method | CHEMISTRY, MULTIDISCIPLINARY | FAMILY | Error analysis (Mathematics) | Analysis | Methods

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 04/2019, Volume 42, Issue 6, pp. 1955 - 1966

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and...

periodic or oscillatory behavior | trigonometrical fitting | two‐derivative Runge‐Kutta methods | two-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | INTEGRATION | STABILITY | INITIAL-VALUE PROBLEMS | Coefficients | Test procedures | Construction methods

periodic or oscillatory behavior | trigonometrical fitting | two‐derivative Runge‐Kutta methods | two-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | INTEGRATION | STABILITY | INITIAL-VALUE PROBLEMS | Coefficients | Test procedures | Construction methods

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 08/2019, Volume 142, pp. 171 - 189

A new family of modified two-derivative Runge-Kutta-Nyström (TDRKN) methods are proposed for solving initial value problems of second-order oscillatory...

Two-derivative Runge-Kutta-Nyström method | Nyström tree | Order condition | Periodicity region | B-series | Trigonometrical fitting | Two-derivative Runge-Kutta-Nystrom method | MATHEMATICS, APPLIED | Nystrom tree | SYSTEMS | Analysis | Methods | Differential equations

Two-derivative Runge-Kutta-Nyström method | Nyström tree | Order condition | Periodicity region | B-series | Trigonometrical fitting | Two-derivative Runge-Kutta-Nystrom method | MATHEMATICS, APPLIED | Nystrom tree | SYSTEMS | Analysis | Methods | Differential equations

Journal Article

Journal of Mathematical Chemistry, ISSN 0259-9791, 8/2018, Volume 56, Issue 7, pp. 1924 - 1934

Optimized explicit two-derivative Runge–Kutta methods with increased phase-lag and dissipation order for the numerical integration of the Schrödinger equation...

Theoretical and Computational Chemistry | Chemistry | Physical Chemistry | Dissipation | Two-derivative Runge–Kutta method | Phase-lag | Math. Applications in Chemistry | Schrödinger equation | INFINITY | NUMERICAL-SOLUTION | IVPS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | MULTISTEP METHODS | Two-derivative Runge-Kutta method | OSCILLATING SOLUTIONS | CHEMISTRY, MULTIDISCIPLINARY | Research | Mathematical research | Iterative methods (Mathematics)

Theoretical and Computational Chemistry | Chemistry | Physical Chemistry | Dissipation | Two-derivative Runge–Kutta method | Phase-lag | Math. Applications in Chemistry | Schrödinger equation | INFINITY | NUMERICAL-SOLUTION | IVPS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | MULTISTEP METHODS | Two-derivative Runge-Kutta method | OSCILLATING SOLUTIONS | CHEMISTRY, MULTIDISCIPLINARY | Research | Mathematical research | Iterative methods (Mathematics)

Journal Article

Numerical Algorithms, ISSN 1017-1398, 3/2010, Volume 53, Issue 2, pp. 171 - 194

The theory of Runge-Kutta methods for problems of the form y′ = f(y) is extended to include the second derivative y′′ = g(y): = f′(y)f(y). We present an...

65L05 | Explicit methods | Mildly stiff problems | Rooted trees | Numeric Computing | Pseudo stage order | Order conditions | Theory of Computation | Stability region | Algebra | Algorithms | Computer Science | Two-derivative Runge-Kutta methods | Mathematics, general | Stage order | Rooted trees | Mildly stiff problems | ORDER | MATHEMATICS, APPLIED | INTEGRATION METHODS | Numerical analysis | Computation | Derivation | Mathematical models | Runge-Kutta method | Derivatives | Standards

65L05 | Explicit methods | Mildly stiff problems | Rooted trees | Numeric Computing | Pseudo stage order | Order conditions | Theory of Computation | Stability region | Algebra | Algorithms | Computer Science | Two-derivative Runge-Kutta methods | Mathematics, general | Stage order | Rooted trees | Mildly stiff problems | ORDER | MATHEMATICS, APPLIED | INTEGRATION METHODS | Numerical analysis | Computation | Derivation | Mathematical models | Runge-Kutta method | Derivatives | Standards

Journal Article

Numerical Algorithms, ISSN 1017-1398, 1/2017, Volume 74, Issue 1, pp. 247 - 265

We introduce an algorithm for a numerical integration of ordinary differential equations in the form of y′ = f(y). We extend the two-derivative Runge-Kutta...

Algorithms | Algebra | Rooted trees | Numerical Analysis | Computer Science | Numeric Computing | Two-derivative Runge-Kutta methods | Order conditions | Theory of Computation | Multi-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | INTEGRATION METHODS | ORDINARY DIFFERENTIAL-EQUATIONS | Analysis | Methods | Differential equations

Algorithms | Algebra | Rooted trees | Numerical Analysis | Computer Science | Numeric Computing | Two-derivative Runge-Kutta methods | Order conditions | Theory of Computation | Multi-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | INTEGRATION METHODS | ORDINARY DIFFERENTIAL-EQUATIONS | Analysis | Methods | Differential equations

Journal Article

Numerical Algorithms, ISSN 1017-1398, 3/2014, Volume 65, Issue 3, pp. 687 - 703

We develop a novel and general approach to the discretization of partial differential equations. This approach overcomes the rigid restriction of the...

Algorithms | Algebra | Advection equation | Heat equation | Two-derivative Runge–Kutta methods | Numerical Analysis | PDE methods | Computer Science | Numeric Computing | Theory of Computation | Stability region | Two-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | SCHEMES | Advection | Discretization | Partial differential equations | Mathematical analysis | Mathematical models | Runge-Kutta method | Derivatives | Flexibility

Algorithms | Algebra | Advection equation | Heat equation | Two-derivative Runge–Kutta methods | Numerical Analysis | PDE methods | Computer Science | Numeric Computing | Theory of Computation | Stability region | Two-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | SCHEMES | Advection | Discretization | Partial differential equations | Mathematical analysis | Mathematical models | Runge-Kutta method | Derivatives | Flexibility

Journal Article

Numerical Algorithms, ISSN 1017-1398, 12/2015, Volume 70, Issue 4, pp. 897 - 927

Classical Runge-Kutta-Nyström (RKN) methods for second-order ordinary differential equations are extended to two-derivative Runge-Kutta-Nyström (TDRKN) methods...

65L05 | 65L06 | Nyström tree theory | Numeric Computing | Order conditions | Theory of Computation | Second-order ordinary differential equations | Algorithms | Algebra | Numerical Analysis | Two-derivative Runge-Kutta-Nystöm methods | Computer Science | 65N40 | 65M20 | MATHEMATICS, APPLIED | Two-derivative Runge-Kutta-Nystom methods | 2ND-DERIVATIVE MULTISTEP METHODS | NUMERICAL-INTEGRATION | Nystrom tree theory | Analysis | Methods | Differential equations | Construction | Numerical analysis | Stability | Mathematical models | Runge-Kutta method | Derivatives

65L05 | 65L06 | Nyström tree theory | Numeric Computing | Order conditions | Theory of Computation | Second-order ordinary differential equations | Algorithms | Algebra | Numerical Analysis | Two-derivative Runge-Kutta-Nystöm methods | Computer Science | 65N40 | 65M20 | MATHEMATICS, APPLIED | Two-derivative Runge-Kutta-Nystom methods | 2ND-DERIVATIVE MULTISTEP METHODS | NUMERICAL-INTEGRATION | Nystrom tree theory | Analysis | Methods | Differential equations | Construction | Numerical analysis | Stability | Mathematical models | Runge-Kutta method | Derivatives

Journal Article

International Journal of Modern Physics C, ISSN 0129-1831, 10/2013, Volume 24, Issue 10, p. 1350073

Two exponentially fitted two-derivative Runge–Kutta (EFTDRK) methods of algebraic order four are derived. The asymptotic expressions of the local errors for...

Two-derivative Rung-Kutta method | exponential fitting | Woods-Saxon potential | error analysis | Schrödinger equation | 2ND-ORDER IVPS | SYMPLECTIC METHODS | PHYSICS, MATHEMATICAL | INITIAL-VALUE PROBLEMS | PHASE-LAG | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | ORDER INFINITY | INTEGRATION | CONSTRUCTION | OSCILLATING SOLUTIONS | EXPLICIT

Two-derivative Rung-Kutta method | exponential fitting | Woods-Saxon potential | error analysis | Schrödinger equation | 2ND-ORDER IVPS | SYMPLECTIC METHODS | PHYSICS, MATHEMATICAL | INITIAL-VALUE PROBLEMS | PHASE-LAG | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | ORDER INFINITY | INTEGRATION | CONSTRUCTION | OSCILLATING SOLUTIONS | EXPLICIT

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 2011, Volume 61, Issue 10, pp. 1046 - 1058

An extension of general linear methods (GLMs), so-called SGLMs (GLMs with second derivative), was introduced to the case in which second derivatives, as well...

Runge–Kutta stability | Ordinary differential equation | Two-derivative methods | General linear methods | A-stability | Runge-Kutta stability | MATHEMATICS, APPLIED | STIFF SYSTEMS | CONSTRUCTION | MULTISTAGE INTEGRATION METHODS | ORDINARY DIFFERENTIAL-EQUATIONS | EXPLICIT | Architecture | Methods

Runge–Kutta stability | Ordinary differential equation | Two-derivative methods | General linear methods | A-stability | Runge-Kutta stability | MATHEMATICS, APPLIED | STIFF SYSTEMS | CONSTRUCTION | MULTISTAGE INTEGRATION METHODS | ORDINARY DIFFERENTIAL-EQUATIONS | EXPLICIT | Architecture | Methods

Journal Article

Journal of Mathematical Chemistry, ISSN 0259-9791, 1/2014, Volume 52, Issue 1, pp. 240 - 254

This paper focuses on adapted two-derivative Runge-Kutta (TDRK) type methods for solving the Schrödinger equation. Two new TDRK methods are derived by...

Theoretical and Computational Chemistry | Chemistry | Phase fitting | Error analysis | Physical Chemistry | Two-derivative Runge-Kutta methods | Math. Applications in Chemistry | Schrödinger equation | FITTING BDF ALGORITHMS | STABILITY | CHEMISTRY, MULTIDISCIPLINARY | FAMILY | PHASE-LAG | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | ORDER INFINITY | MULTISTEP METHODS | INTEGRATION | SCIENCE | EXPLICIT | Numerical analysis | Research | Error analysis (Mathematics)

Theoretical and Computational Chemistry | Chemistry | Phase fitting | Error analysis | Physical Chemistry | Two-derivative Runge-Kutta methods | Math. Applications in Chemistry | Schrödinger equation | FITTING BDF ALGORITHMS | STABILITY | CHEMISTRY, MULTIDISCIPLINARY | FAMILY | PHASE-LAG | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Schrodinger equation | ORDER INFINITY | MULTISTEP METHODS | INTEGRATION | SCIENCE | EXPLICIT | Numerical analysis | Research | Error analysis (Mathematics)

Journal Article

Numerical Algorithms, ISSN 1017-1398, 3/2014, Volume 65, Issue 3, pp. 651 - 667

Explicit trigonometrically fitted two-derivative Runge-Kutta (TFTDRK) methods solving second-order differential equations with oscillatory solutions are...

Algorithms | Algebra | Two-derivative Runge-Kutta method | Numerical Analysis | Computer Science | Numeric Computing | Oscillatory problem | Theory of Computation | Stability analysis | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MULTISTEP METHODS | NYSTROM METHODS | STABILITY | SCHRODINGER-EQUATION | Analysis | Methods | Differential equations | Construction | Equivalence | Mathematical analysis | Mathematical models | Runge-Kutta method | Robustness | Derivatives

Algorithms | Algebra | Two-derivative Runge-Kutta method | Numerical Analysis | Computer Science | Numeric Computing | Oscillatory problem | Theory of Computation | Stability analysis | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MULTISTEP METHODS | NYSTROM METHODS | STABILITY | SCHRODINGER-EQUATION | Analysis | Methods | Differential equations | Construction | Equivalence | Mathematical analysis | Mathematical models | Runge-Kutta method | Robustness | Derivatives

Journal Article

Journal of the Nigerian Mathematical Society, ISSN 0189-8965, 08/2015, Volume 34, Issue 2, pp. 128 - 142

We introduce a new class of implicit two-derivative Runge–Kutta collocation methods designed for the numerical solution of systems of equations and show how...

System of equations | Block hybrid discrete scheme | Continuous scheme | Two-derivative Runge–Kutta methods

System of equations | Block hybrid discrete scheme | Continuous scheme | Two-derivative Runge–Kutta methods

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 02/2020, Volume 43, Issue 3, pp. 1267 - 1277

In this work, we consider two‐derivative Runge‐Kutta methods for the numerical integration of first‐order differential equations with oscillatory solution. We...

phase lag | two‐derivative Runge‐Kutta methods | amplification error | two-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | ORDER INFINITY | LAG | INTEGRATION | PAIRS | Amplification | Numerical integration | Numerical methods | Differential equations

phase lag | two‐derivative Runge‐Kutta methods | amplification error | two-derivative Runge-Kutta methods | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | ORDER INFINITY | LAG | INTEGRATION | PAIRS | Amplification | Numerical integration | Numerical methods | Differential equations

Journal Article

AIP Conference Proceedings, ISSN 0094-243X, 11/2018, Volume 2040, Issue 1

In this work we consider Two Derivative Runge-Kutta methods for the numerical integration of oscillatory problems. We compare numerical methods with constant...

trigonometrically fitted methods | Two Derivative Runge Kutta methods | Numerical methods | Runge-Kutta method | Numerical integration

trigonometrically fitted methods | Two Derivative Runge Kutta methods | Numerical methods | Runge-Kutta method | Numerical integration

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 02/2014, Volume 76, pp. 1 - 18

Second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) have been...

A- and L-stability | Order conditions | Runge–Kutta stability | Ordinary differential equation | Two-derivative methods | General linear methods | Runge-Kutta stability | ORDER | MATHEMATICS, APPLIED | APPROXIMATIONS | STABILITY | ORDINARY DIFFERENTIAL-EQUATIONS | Architecture | Methods | Multistage | Construction | Stability | Architecture (computers) | Mathematical models | Runge-Kutta method | Derivatives

A- and L-stability | Order conditions | Runge–Kutta stability | Ordinary differential equation | Two-derivative methods | General linear methods | Runge-Kutta stability | ORDER | MATHEMATICS, APPLIED | APPROXIMATIONS | STABILITY | ORDINARY DIFFERENTIAL-EQUATIONS | Architecture | Methods | Multistage | Construction | Stability | Architecture (computers) | Mathematical models | Runge-Kutta method | Derivatives

Journal Article

AIP Conference Proceedings, ISSN 0094-243X, 07/2018, Volume 1978, Issue 1

In this work we consider Two Derivative Runge-Kutta methods for the numerical integration of oscillatory problems. We construct methods with variable...

phase lag | amplification error | Two Derivative Runge Kutta methods | Numerical methods | Runge-Kutta method | Numerical integration

phase lag | amplification error | Two Derivative Runge Kutta methods | Numerical methods | Runge-Kutta method | Numerical integration

Journal Article

AIP Conference Proceedings, ISSN 0094-243X, 07/2017, Volume 1863, Issue 1

In this work we consider Two Derivative Runge-Kutta methods. We present the order conditions after applying simplifying conditions. Two new methods of...

order conditions | Two Derivative Runge Kutta methods | Two body problem | Runge-Kutta method

order conditions | Two Derivative Runge Kutta methods | Two body problem | Runge-Kutta method

Journal Article

Computational and Applied Mathematics, ISSN 0101-8205, 11/2018, Volume 37, Issue 5, pp. 6920 - 6954

We introduce a class of methods for the numerical solution of ordinary differential equations. These methods called as two-derivative two-step Runge–Kutta...

65L05 | 65L06 | Computational Mathematics and Numerical Analysis | Two-derivative Runge–Kutta methods | Order conditions | Mathematics | Two-step Runge–Kutta methods | Mathematical Applications in Computer Science | Rooted trees | Second-derivative general linear methods | 65L20 | Applications of Mathematics | Mathematical Applications in the Physical Sciences | MATHEMATICS, APPLIED | Two-step Runge-Kutta methods | INTEGRATION | CONSTRUCTION | Two-derivative Runge-Kutta methods | 2ND-DERIVATIVE MULTISTEP METHODS | IMPLEMENTATION | FORMULAS

65L05 | 65L06 | Computational Mathematics and Numerical Analysis | Two-derivative Runge–Kutta methods | Order conditions | Mathematics | Two-step Runge–Kutta methods | Mathematical Applications in Computer Science | Rooted trees | Second-derivative general linear methods | 65L20 | Applications of Mathematics | Mathematical Applications in the Physical Sciences | MATHEMATICS, APPLIED | Two-step Runge-Kutta methods | INTEGRATION | CONSTRUCTION | Two-derivative Runge-Kutta methods | 2ND-DERIVATIVE MULTISTEP METHODS | IMPLEMENTATION | FORMULAS

Journal Article

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