The equivalence of stability and convergence for finite difference schemes with singular coefficients

1967 ◽  
Vol 10 (1) ◽  
pp. 20-29 ◽  
Author(s):  
Dennis Eisen
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Singular Coefficients

Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jinye Shen ◽  
Martin Stynes ◽  
Zhi-Zhong Sun
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Mesh Size ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Initial Time ◽  
Finite Difference Schemes

Abstract A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is ( 0 , 1 ) d (0,1)^{d} for some d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t = 0 t=0 : while 𝑢 and u t u_{t} are continuous at t = 0 t=0 , the second-order derivative u t ⁢ t u_{tt} blows up at t = 0 t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v := u t v:=u_{t} ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.

scholarly journals Finite Difference Schemes for Partial Differential Equations with Weak Solutions and Irregular Coefficients

2004 ◽  
Vol 4 (1) ◽  
pp. 48-65 ◽  
Author(s):  
Boško S. Jovanović
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Sobolev Space ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Difference Schemes ◽  
Generalized Solutions ◽  
Finite Difference Schemes ◽  
Singular Coefficients

AbstractA survey of the results concerning the convergence of finite difference schemes for boundary value problems with generalized solutions from the Sobolev space is presented. In particular, difference schemes for some problems with singular coefficients are investigated.

scholarly journals Conservative finite difference schemes for the modified Camassa-Holm equation

JSIAM Letters ◽  
2011 ◽  
Vol 3 (0) ◽  
pp. 37-40 ◽  
Author(s):  
Yuto Miyatake ◽  
Takayasu Matsuo ◽  
Daisuke Furihata
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Holm Equation
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