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We present high-order compact schemes for a linear second-order parabolicpartial differential equation (PDE) with mixed second-order derivative terms intwo spatial dimensions. The schemes are applied to option pricing PDE for afamily of stochastic volatility models. We use a non-uniform grid with moregrid-points around the strike price. The schemes are fourth-order accurate inspace and second-order accurate in time for vanishing correlation. In ournumerical convergence study we achieve fourth-order accuracy also for non-zerocorrelation. A combination of Crank-Nicolson and BDF-4 discretisation isapplied in time. Numerical examples confirm that a standard, second-orderfinite difference scheme is significantly outperformed.

Essentially high-order compact schemes with application to stochastic volatility models on non-uniform grids