A three‐level linearized finite difference scheme for the camassa–holm equation

Cao, Hai-Yan; Sun, Zhi‐zhong; Gao, Guang‐Hua

doi:10.1002/num.21819

Cited by 9 publications

(5 citation statements)

References 31 publications

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“…Crank-Nicolson Scheme and Richardson's Extrapolation. We use a three-level Crank-Nicolson scheme (see [18][19][20][21]) of second-order accuracy to solve the nonlinear and inhomogeneous partial differential equation given by (19) and use Richardson's extrapolation technique for further improving accuracy. Numerically, one can only solve (19) over a finite domain…”

Section: Methodsmentioning

confidence: 99%

“…is discretization scheme leads to a set of linear equations. Based on the expressions given by (21) and (22), equation (19) can be discretized as…”

Section: Methodsmentioning

confidence: 99%

“…e method given by (21) and 22is a two-step method, namely, f n+1 depends on f n and f n− 1 . At the zeroth step, f 0 is given by initial condition (16), namely, f 0 m � (1 − α) (1/c) , for 0 ≤ m ≤ M. We now determine f 1 , the solution at the first step.…”

Section: Methodsmentioning

confidence: 99%

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Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility

Zhang²

2020

Complexity

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Based on the method of dynamic programming, this paper uses analysis methods governed by the nonlinear and inhomogeneous partial differential equation to study modern portfolio management problems with stochastic volatility, incomplete markets, limited investment scope, and constant relative risk aversion (CRRA). In this paper, a three-level Crank–Nicolson finite difference scheme is used to determine numerical solutions under this general setting. One of the main contributions of this paper is to apply this three-level technology to solve the portfolio selection problem. In addition, we have used a technique to deal with the nonlinear term, which is another novelty in performing the Crank–Nicolson algorithm. The Crank–Nicolson algorithm has also been extended to third-order accuracy by performing Richardson’s extrapolation. The accuracy of the proposed algorithm is much higher than the traditional finite difference method. Lastly, experiments are conducted to show the performance of the proposed algorithm.

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Section: Methodsmentioning

confidence: 99%

“…is discretization scheme leads to a set of linear equations. Based on the expressions given by (21) and (22), equation (19) can be discretized as…”

Section: Methodsmentioning

confidence: 99%

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Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility

Zhang²

2020

Complexity

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“…

δ_{t} (u_{j}^{n}) + \frac{α}{m + 1} δ_{\overset{x}{true}} ({((), u_{j}^{n})}^{m} {\overset{u}{true}}_{j}^{n}) + \frac{β}{6} δ_{x \overset{true‾}{x} \overset{true^}{x}}^{(3)} (u_{j}^{n} {\overset{u}{true}}_{j}^{n}) + (γ - \frac{β}{3}) u_{j}^{n} δ_{x \overset{true‾}{x} \overset{true^}{x}}^{(3)} ({\overset{true‾}{u}}_{j}^{n}) + {μ δ}_{xx \overset{true‾}{x} \overset{true‾}{x} \overset{true^}{x}}^{(5)} ({\overset{true‾}{u}}_{j}^{n}) = 0 .

Another approach for solving Equation (1), based on the existing finite difference schemes for solving the Camassa–Holm equation [22], Equation (1) can be rewritten as

u_{t} + \frac{α}{m + 1} {((), u^{m + 1})}_{x} + false(β - γ false) u_{italicxx} u_{x} + γ {((), {italicuu}_{italicxx})}_{x} + μ u_{italicxxxxx} = 0,

where the formula

{italicuu}_{italicxxx} = {((), {italicuu}_{italicxx})}_{x}

…”

Section: Finite Difference Schemementioning

confidence: 99%

“…Another approach for solving Equation (1), based on the existing finite difference schemes for solving the Camassa-Holm equation [22], Equation (1) can be rewritten as…”

Section: Construction Of Difference Schemesmentioning

confidence: 99%

A novel convenient finite difference method for shallow water waves derived by fifth‐order Kortweg and De‐Vries‐type equation

Poochinapan

Wongsaijai

2022

Numerical Methods Partial

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In this research, two new finite difference schemes are derived and presented for estimating a solution to the fifth-order Kortweg and De-Vries equation. The global conservation law on any time-space regions which yields a three-level linear-implicit algorithm with a diagonal system is exactly preserved by the intended finite difference method. Theoretically verified and numerically proved, the created schemes are unconditionally stable and have the second-order accuracy both in time and space. The obtained results guarantee that the novel idea offers a new aspect to analyze the wave behavior.

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Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation

2018

Commun. Math. Stat.

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A three‐level linearized finite difference scheme for the camassa–holm equation

Cited by 9 publications

References 31 publications

Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility

Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility

A novel convenient finite difference method for shallow water waves derived by fifth‐order Kortweg and De‐Vries‐type equation

Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation

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