Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  present a high-order compact finite difference approach for a class of parabolic partial differential equations with time- and space-dependent coefficients as ll as with mixed second-order derivative terms in $n$ spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance.  derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover,  perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example  consider the pricing of European por put basket options in the multidimensional Black--Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions  employ th...
['Bertram Düring', 'Christof Heuer']
I am writing a research paper on a idea High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions
[{"from": "human", "value": "I am writing a research paper on a idea High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  present a high-order compact finite difference approach for a class of parabolic partial differential equations with time- and space-dependent coefficients as ll as with mixed second-order derivative terms in $n$ spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance.  derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover,  perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example  consider the pricing of European por put basket options in the multidimensional Black--Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions  employ th..."}]