📐 ODE Solver Stability Regions
Visualize stability regions in the complex plane for popular ODE integrators (Runge-Kutta, Euler, etc.).
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
● ACTIVE
📥 Time-Stepping Scheme & Test Eigenvalue
📖 Stability Domain $|G(z)| \le 1$:
$$\text{Test Equation: } \frac{du}{dt} = \lambda u \implies u^{n+1} = G(\lambda \Delta t)\, u^n$$
Amplification Factor:
$$G(z) = 1 + z + z^2/2 + z^3/6 + z^4/24$$
$$G(z) = 1 + z + z^2/2 + z^3/6 + z^4/24$$
Re(z) intercept: [-2.785, 0]. Imaginary intercept: [-2.828*i, +2.828*i]. Ideal for convective CFD waves.
STABLE ✅
Eigenvalue Status
0.1295
Amplification $|G(z)|$
-1.50 + 0.80i
Probe $z = \lambda \Delta t$
Conditional
Global Property
💡
Complex Stability Diagnostic
For Runge-Kutta 4th Order (Classical RK4) at test point $z = \mathbf{-1.5 + 0.8 i}$, the amplification magnitude is $|G(z)| = \mathbf{0.1295}$.
The eigenvalue lies strictly inside the stable region ($|G| \le 1$). Numerical errors will decay exponentially at a rate of 87.0% per time step.
📈 Stability Boundary $|G(z)| = 1$ in Complex $z$-Plane
Shaded Area = Stable Region🔍 View Raw GNU Fortran Double-Precision Solver Output
SCHEME_1=Forward Euler (Explicit) FORMULA_1=G = 1 + z REGION_1=Disk center(-1,0) radius 1 PROBE_1= 9.43398113E-01 STABLE_1=YES CONTOUR_1_START 0.00000000, 0.00000000 -0.00197327, 0.06279052 -0.00788530, 0.12533323 -0.01771275, 0.18738131 -0.03141684, 0.24868989 -0.04894348, 0.30901699 -0.07022351, 0.36812455 -0.09517295, 0.42577929 -0.12369332, 0.48175367 -0.15567207, 0.53582679 -0.19098301, 0.58778525 -0.22948676, 0.63742399 -0.27103137, 0.68454711 -0.31545289, 0.72896863 -0.36257601, 0.77051324 -0.41221475, 0.80901699 -0.46417321, 0.84432793 -0.51824633, 0.87630668 -0.57422071, 0.90482705 -0.63187545, 0.92977649 -0.69098301, 0.95105652 -0.75131011, 0.96858316 -0.81261869, 0.98228725 -0.87466677, 0.99211470 -0.93720948, 0.99802673 -1.00000000, 1.00000000 -1.06279052, 0.99802673 -1.12533323, 0.99211470 -1.18738131, 0.98228725 -1.24868989, 0.96858316 -1.30901699, 0.95105652 -1.36812455, 0.92977649 -1.42577929, 0.90482705 -1.48175367, 0.87630668 -1.53582679, 0.84432793 -1.58778525, 0.80901699 -1.63742399, 0.77051324 -1.68454711, 0.72896863 -1.72896863, 0.68454711 -1.77051324, 0.63742399 -1.80901699, 0.58778525 -1.84432793, 0.53582679 -1.87630668, 0.48175367 -1.90482705, 0.42577929 -1.92977649, 0.36812455 -1.95105652, 0.30901699 -1.96858316, 0.24868989 -1.98228725, 0.18738131 -1.99211470, 0.12533323 -1.99802673, 0.06279052 -2.00000000, -0.00000000 -1.99802673, -0.06279052 -1.99211470, -0.12533323 -1.98228725, -0.18738131 -1.96858316, -0.24868989 -1.95105652, -0.30901699 -1.92977649, -0.36812455 -1.90482705, -0.42577929 -1.87630668, -0.48175367 -1.84432793, -0.53582679 -1.80901699, -0.58778525 -1.77051324, -0.63742399 -1.72896863, -0.68454711 -1.68454711, -0.72896863 -1.63742399, -0.77051324 -1.58778525, -0.80901699 -1.53582679, -0.84432793 -1.48175367, -0.87630668 -1.42577929, -0.90482705 -1.36812455, -0.92977649 -1.30901699, -0.95105652 -1.24868989, -0.96858316 -1.18738131, -0.98228725 -1.12533323, -0.99211470 -1.06279052, -0.99802673 -1.00000000, -1.00000000 -0.93720948, -0.99802673 -0.87466677, -0.99211470 -0.81261869, -0.98228725 -0.75131011, -0.96858316 -0.69098301, -0.95105652 -0.63187545, -0.92977649 -0.57422071, -0.90482705 -0.51824633, -0.87630668 -0.46417321, -0.84432793 -0.41221475, -0.80901699 -0.36257601, -0.77051324 -0.31545289, -0.72896863 -0.27103137, -0.68454711 -0.22948676, -0.63742399 -0.19098301, -0.58778525 -0.15567207, -0.53582679 -0.12369332, -0.48175367 -0.09517295, -0.42577929 -0.07022351, -0.36812455 -0.04894348, -0.30901699 -0.03141684, -0.24868989 -0.01771275, -0.18738131 -0.00788530, -0.12533323 -0.00197327, -0.06279052 0.00000000, 0.00000000 CONTOUR_1_END SCHEME_2=Backward Euler (Implicit) FORMULA_2=G = 1 / (1 - z) REGION_2=Exterior of disk center(1,0) radius 1 PROBE_2= 3.80969659E-01 STABLE_2=YES CONTOUR_2_START 2.00000000, 0.00000000 1.99802673, 0.06279052 1.99211470, 0.12533323 1.98228725, 0.18738131 1.96858316, 0.24868989 1.95105652, 0.30901699 1.92977649, 0.36812455 1.90482705, 0.42577929 1.87630668, 0.48175367 1.84432793, 0.53582679 1.80901699, 0.58778525 1.77051324, 0.63742399 1.72896863, 0.68454711 1.68454711, 0.72896863 1.63742399, 0.77051324 1.58778525, 0.80901699 1.53582679, 0.84432793 1.48175367, 0.87630668 1.42577929, 0.90482705 1.36812455, 0.92977649 1.30901699, 0.95105652 1.24868989, 0.96858316 1.18738131, 0.98228725 1.12533323, 0.99211470 1.06279052, 0.99802673 1.00000000, 1.00000000 0.93720948, 0.99802673 0.87466677, 0.99211470 0.81261869, 0.98228725 0.75131011, 0.96858316 0.69098301, 0.95105652 0.63187545, 0.92977649 0.57422071, 0.90482705 0.51824633, 0.87630668 0.46417321, 0.84432793 0.41221475, 0.80901699 0.36257601, 0.77051324 0.31545289, 0.72896863 0.27103137, 0.68454711 0.22948676, 0.63742399 0.19098301, 0.58778525 0.15567207, 0.53582679 0.12369332, 0.48175367 0.09517295, 0.42577929 0.07022351, 0.36812455 0.04894348, 0.30901699 0.03141684, 0.24868989 0.01771275, 0.18738131 0.00788530, 0.12533323 0.00197327, 0.06279052 0.00000000, -0.00000000 0.00197327, -0.06279052 0.00788530, -0.12533323 0.01771275, -0.18738131 0.03141684, -0.24868989 0.04894348, -0.30901699 0.07022351, -0.36812455 0.09517295, -0.42577929 0.12369332, -0.48175367 0.15567207, -0.53582679 0.19098301, -0.58778525 0.22948676, -0.63742399 0.27103137, -0.68454711 0.31545289, -0.72896863 0.36257601, -0.77051324 0.41221475, -0.80901699 0.46417321, -0.84432793 0.51824633, -0.87630668 0.57422071, -0.90482705 0.63187545, -0.92977649 0.69098301, -0.95105652 0.75131011, -0.96858316 0.81261869, -0.98228725 0.87466677, -0.99211470 0.93720948, -0.99802673 1.00000000, -1.00000000 1.06279052, -0.99802673 1.12533323, -0.99211470 1.18738131, -0.98228725 1.24868989, -0.96858316 1.30901699, -0.95105652 1.36812455, -0.92977649 1.42577929, -0.90482705 1.48175367, -0.87630668 1.53582679, -0.84432793 1.58778525, -0.80901699 1.63742399, -0.77051324 1.68454711, -0.72896863 1.72896863, -0.68454711 1.77051324, -0.63742399 1.80901699, -0.58778525 1.84432793, -0.53582679 1.87630668, -0.48175367 1.90482705, -0.42577929 1.92977649, -0.36812455 1.95105652, -0.30901699 1.96858316, -0.24868989 1.98228725, -0.18738131 1.99211470, -0.12533323 1.99802673, -0.06279052 2.00000000, 0.00000000 CONTOUR_2_END SCHEME_3=Crank-Nicolson FORMULA_3=G = (2 + z) / (2 - z) REGION_3=Left half-plane Re(z) <= 0 PROBE_3= 2.62765622E-01 STABLE_3=YES CONTOUR_3_START 0.00000000, -4.00000000 0.00000000, -3.92000000 0.00000000, -3.84000000 0.00000000, -3.76000000 0.00000000, -3.68000000 0.00000000, -3.60000000 0.00000000, -3.52000000 0.00000000, -3.44000000 0.00000000, -3.36000000 0.00000000, -3.28000000 0.00000000, -3.20000000 0.00000000, -3.12000000 0.00000000, -3.04000000 0.00000000, -2.96000000 0.00000000, -2.88000000 0.00000000, -2.80000000 0.00000000, -2.72000000 0.00000000, -2.64000000 0.00000000, -2.56000000 0.00000000, -2.48000000 0.00000000, -2.40000000 0.00000000, -2.32000000 0.00000000, -2.24000000 0.00000000, -2.16000000 0.00000000, -2.08000000 0.00000000, -2.00000000 0.00000000, -1.92000000 0.00000000, -1.84000000 0.00000000, -1.76000000 0.00000000, -1.68000000 0.00000000, -1.60000000 0.00000000, -1.52000000 0.00000000, -1.44000000 0.00000000, -1.36000000 0.00000000, -1.28000000 0.00000000, -1.20000000 0.00000000, -1.12000000 0.00000000, -1.04000000 0.00000000, -0.96000000 0.00000000, -0.88000000 0.00000000, -0.80000000 0.00000000, -0.72000000 0.00000000, -0.64000000 0.00000000, -0.56000000 0.00000000, -0.48000000 0.00000000, -0.40000000 0.00000000, -0.32000000 0.00000000, -0.24000000 0.00000000, -0.16000000 0.00000000, -0.08000000 0.00000000, 0.00000000 0.00000000, 0.08000000 0.00000000, 0.16000000 0.00000000, 0.24000000 0.00000000, 0.32000000 0.00000000, 0.40000000 0.00000000, 0.48000000 0.00000000, 0.56000000 0.00000000, 0.64000000 0.00000000, 0.72000000 0.00000000, 0.80000000 0.00000000, 0.88000000 0.00000000, 0.96000000 0.00000000, 1.04000000 0.00000000, 1.12000000 0.00000000, 1.20000000 0.00000000, 1.28000000 0.00000000, 1.36000000 0.00000000, 1.44000000 0.00000000, 1.52000000 0.00000000, 1.60000000 0.00000000, 1.68000000 0.00000000, 1.76000000 0.00000000, 1.84000000 0.00000000, 1.92000000 0.00000000, 2.00000000 0.00000000, 2.08000000 0.00000000, 2.16000000 0.00000000, 2.24000000 0.00000000, 2.32000000 0.00000000, 2.40000000 0.00000000, 2.48000000 0.00000000, 2.56000000 0.00000000, 2.64000000 0.00000000, 2.72000000 0.00000000, 2.80000000 0.00000000, 2.88000000 0.00000000, 2.96000000 0.00000000, 3.04000000 0.00000000, 3.12000000 0.00000000, 3.20000000 0.00000000, 3.28000000 0.00000000, 3.36000000 0.00000000, 3.44000000 0.00000000, 3.52000000 0.00000000, 3.60000000 0.00000000, 3.68000000 0.00000000, 3.76000000 0.00000000, 3.84000000 0.00000000, 3.92000000 0.00000000, 4.00000000 CONTOUR_3_END SCHEME_4=Leapfrog (Midpoint) FORMULA_4=G = z +/- sqrt(z^2 + 1) REGION_4=Imaginary segment [-i, i] PROBE_4= 3.57676744E+00 STABLE_4=NO CONTOUR_4_START 0.00000000, -1.00000000 0.00000000, -0.98000000 0.00000000, -0.96000000 0.00000000, -0.94000000 0.00000000, -0.92000000 0.00000000, -0.90000000 0.00000000, -0.88000000 0.00000000, -0.86000000 0.00000000, -0.84000000 0.00000000, -0.82000000 0.00000000, -0.80000000 0.00000000, -0.78000000 0.00000000, -0.76000000 0.00000000, -0.74000000 0.00000000, -0.72000000 0.00000000, -0.70000000 0.00000000, -0.68000000 0.00000000, -0.66000000 0.00000000, -0.64000000 0.00000000, -0.62000000 0.00000000, -0.60000000 0.00000000, -0.58000000 0.00000000, -0.56000000 0.00000000, -0.54000000 0.00000000, -0.52000000 0.00000000, -0.50000000 0.00000000, -0.48000000 0.00000000, -0.46000000 0.00000000, -0.44000000 0.00000000, -0.42000000 0.00000000, -0.40000000 0.00000000, -0.38000000 0.00000000, -0.36000000 0.00000000, -0.34000000 0.00000000, -0.32000000 0.00000000, -0.30000000 0.00000000, -0.28000000 0.00000000, -0.26000000 0.00000000, -0.24000000 0.00000000, -0.22000000 0.00000000, -0.20000000 0.00000000, -0.18000000 0.00000000, -0.16000000 0.00000000, -0.14000000 0.00000000, -0.12000000 0.00000000, -0.10000000 0.00000000, -0.08000000 0.00000000, -0.06000000 0.00000000, -0.04000000 0.00000000, -0.02000000 0.00000000, 0.00000000 0.00000000, 0.02000000 0.00000000, 0.04000000 0.00000000, 0.06000000 0.00000000, 0.08000000 0.00000000, 0.10000000 0.00000000, 0.12000000 0.00000000, 0.14000000 0.00000000, 0.16000000 0.00000000, 0.18000000 0.00000000, 0.20000000 0.00000000, 0.22000000 0.00000000, 0.24000000 0.00000000, 0.26000000 0.00000000, 0.28000000 0.00000000, 0.30000000 0.00000000, 0.32000000 0.00000000, 0.34000000 0.00000000, 0.36000000 0.00000000, 0.38000000 0.00000000, 0.40000000 0.00000000, 0.42000000 0.00000000, 0.44000000 0.00000000, 0.46000000 0.00000000, 0.48000000 0.00000000, 0.50000000 0.00000000, 0.52000000 0.00000000, 0.54000000 0.00000000, 0.56000000 0.00000000, 0.58000000 0.00000000, 0.60000000 0.00000000, 0.62000000 0.00000000, 0.64000000 0.00000000, 0.66000000 0.00000000, 0.68000000 0.00000000, 0.70000000 0.00000000, 0.72000000 0.00000000, 0.74000000 0.00000000, 0.76000000 0.00000000, 0.78000000 0.00000000, 0.80000000 0.00000000, 0.82000000 0.00000000, 0.84000000 0.00000000, 0.86000000 0.00000000, 0.88000000 0.00000000, 0.90000000 0.00000000, 0.92000000 0.00000000, 0.94000000 0.00000000, 0.96000000 0.00000000, 0.98000000 0.00000000, 1.00000000 CONTOUR_4_END SCHEME_5=RK2 (Heun) FORMULA_5=G = 1 + z + z^2/2 REGION_5=Closed region (computed numerically) PROBE_5= 5.03015904E-01 STABLE_5=YES CONTOUR_5_START -0.00000000, 0.00000000 -0.00000195, 0.06291454 -0.00003158, 0.12632539 -0.00016249, 0.19072921 -0.00052509, 0.25662154 -0.00131825, 0.32449137 -0.00282520, 0.39480943 -0.00543290, 0.46800775 -0.00965241, 0.54444820 -0.01613613, 0.62437900 -0.02568558, 0.70788086 -0.03924200, 0.79480814 -0.05785266, 0.88473524 -0.08261005, 0.97692107 -0.11456817, 1.07030323 -0.15464753, 1.16352786 -0.20354509, 1.25501212 -0.26166384, 1.34302848 -0.32907087, 1.42579699 -0.40548519, 1.50157300 -0.49029112, 1.56872264 -0.58256989, 1.62578294 -0.68114298, 1.67150706 -0.78462144, 1.70489687 -0.89145802, 1.72522531 -1.00000000, 1.73205081 -1.10854198, 1.72522531 -1.21537856, 1.70489687 -1.31885702, 1.67150706 -1.41743011, 1.62578294 -1.50970888, 1.56872264 -1.59451481, 1.50157300 -1.67092913, 1.42579699 -1.73833616, 1.34302848 -1.79645491, 1.25501212 -1.84535247, 1.16352786 -1.88543183, 1.07030323 -1.91738995, 0.97692107 -1.94214734, 0.88473524 -1.96075800, 0.79480814 -1.97431442, 0.70788086 -1.98386387, 0.62437900 -1.99034759, 0.54444820 -1.99456710, 0.46800775 -1.99717480, 0.39480943 -1.99868175, 0.32449137 -1.99947491, 0.25662154 -1.99983751, 0.19072921 -1.99996842, 0.12632539 -1.99999805, 0.06291454 -2.00000000, -0.00000000 -1.99999805, -0.06291454 -1.99996842, -0.12632539 -1.99983751, -0.19072921 -1.99947491, -0.25662154 -1.99868175, -0.32449137 -1.99717480, -0.39480943 -1.99456710, -0.46800775 -1.99034759, -0.54444820 -1.98386387, -0.62437900 -1.97431442, -0.70788086 -1.96075800, -0.79480814 -1.94214734, -0.88473524 -1.91738995, -0.97692107 -1.88543183, -1.07030323 -1.84535247, -1.16352786 -1.79645491, -1.25501212 -1.73833616, -1.34302848 -1.67092913, -1.42579699 -1.59451481, -1.50157300 -1.50970888, -1.56872264 -1.41743011, -1.62578294 -1.31885702, -1.67150706 -1.21537856, -1.70489687 -1.10854198, -1.72522531 -1.00000000, -1.73205081 -0.89145802, -1.72522531 -0.78462144, -1.70489687 -0.68114298, -1.67150706 -0.58256989, -1.62578294 -0.49029112, -1.56872264 -0.40548519, -1.50157300 -0.32907087, -1.42579699 -0.26166384, -1.34302848 -0.20354509, -1.25501212 -0.15464753, -1.16352786 -0.11456817, -1.07030323 -0.08261005, -0.97692107 -0.05785266, -0.88473524 -0.03924200, -0.79480814 -0.02568558, -0.70788086 -0.01613613, -0.62437900 -0.00965241, -0.54444820 -0.00543290, -0.46800775 -0.00282520, -0.39480943 -0.00131825, -0.32449137 -0.00052509, -0.25662154 -0.00016249, -0.19072921 -0.00003158, -0.12632539 -0.00000195, -0.06291454 -0.00000000, 0.00000000 CONTOUR_5_END SCHEME_6=RK4 (Classical) FORMULA_6=G = 1 + z + z^2/2 + z^3/6 + z^4/24 REGION_6=Closed region (computed numerically) PROBE_6= 1.29530364E-01 STABLE_6=YES CONTOUR_6_START -0.00000000, 0.00000000 0.00000000, 0.06291467 0.00000003, 0.12632938 0.00000033, 0.19076027 0.00000197, 0.25675687 0.00000807, 0.32492232 0.00002626, 0.39593840 0.00007348, 0.47059886 0.00018531, 0.54985653 0.00043447, 0.63489502 0.00096880, 0.72724640 0.00209249, 0.82900301 0.00445166, 0.94324289 0.00950272, 1.07501121 0.02089314, 1.23404782 0.04966853, 1.44474478 0.13661916, 1.79102521 0.23576070, 2.24784037 0.19909886, 2.54821478 0.08736681, 2.74637506 -0.06701557, 2.87143081 -0.24740543, 2.93116231 -0.44175701, 2.92641224 -0.63920380, 2.85599597 -0.82861860, 2.72402933 -1.00000000, 2.55766528 -1.15133152, 2.40534561 -1.28903257, 2.28792839 -1.41912736, 2.19714255 -1.54470870, 2.12150031 -1.66705781, 2.05299283 -1.78646751, 1.98639019 -1.90262792, 1.91818197 -2.01483003, 1.84596897 -2.12210646, 1.76815686 -2.22335272, 1.68380057 -2.31744197, 1.59251378 -2.40333393, 1.49439885 -2.48017428, 1.38997617 -2.54737906, 1.28010329 -2.60469776, 1.16588117 -2.65224805, 1.04854850 -2.69051540, 0.92936870 -2.72031324, 0.80951796 -2.74270525, 0.68998582 -2.75889901, 0.57150093 -2.77012748, 0.45449149 -2.77753694, 0.33908331 -2.78209596, 0.22513107 -2.78453198, 0.11227324 -2.78529356, -0.00000000 -2.78453198, -0.11227324 -2.78209596, -0.22513107 -2.77753694, -0.33908331 -2.77012748, -0.45449149 -2.75889901, -0.57150093 -2.74270525, -0.68998582 -2.72031324, -0.80951796 -2.69051540, -0.92936870 -2.65224805, -1.04854850 -2.60469776, -1.16588117 -2.54737906, -1.28010329 -2.48017428, -1.38997617 -2.40333393, -1.49439885 -2.31744197, -1.59251378 -2.22335272, -1.68380057 -2.12210646, -1.76815686 -2.01483003, -1.84596897 -1.90262792, -1.91818197 -1.78646751, -1.98639019 -1.66705781, -2.05299283 -1.54470870, -2.12150031 -1.41912736, -2.19714255 -1.28903257, -2.28792839 -1.15133152, -2.40534561 -1.00000000, -2.55766528 -0.82861860, -2.72402933 -0.63920380, -2.85599597 -0.44175701, -2.92641224 -0.24740543, -2.93116231 -0.06701557, -2.87143081 0.08736681, -2.74637506 0.19909886, -2.54821478 0.23576070, -2.24784037 0.13661916, -1.79102521 0.04966853, -1.44474478 0.02089314, -1.23404782 0.00950272, -1.07501121 0.00445166, -0.94324289 0.00209249, -0.82900301 0.00096880, -0.72724640 0.00043447, -0.63489502 0.00018531, -0.54985653 0.00007348, -0.47059886 0.00002626, -0.39593840 0.00000807, -0.32492232 0.00000197, -0.25675687 0.00000033, -0.19076027 0.00000003, -0.12632938 0.00000000, -0.06291467 -0.00000000, 0.00000000 CONTOUR_6_END
📘 Calculation Methodology: ODE Time-Integration Stability Regions
Mathematical Model & Theory
For the model test equation $dy/dt = \lambda y$ ($\lambda \in \mathbb{C}$), a time-integration scheme is numerically stable if the amplification factor $R(z)$ ($z = \lambda \Delta t$) lies within the unit disk:
$$|R(z)| \le 1, \quad z = \lambda \Delta t = \text{Re}(z) + i \text{Im}(z)$$
$$\text{RK4: } R(z) = 1 + z + \frac{z^2}{2} + \frac{z^3}{6} + \frac{z^4}{24}, \quad \text{Euler: } R(z) = 1 + z$$
Assumptions
- Semi-discrete ODE system from spatial discretization.
- Eigenvalues of the spatial operator must lie strictly inside the stability domain.
Academic References
- Hairer, E., & Wanner, G.: Solving ODEs I & II, Springer.
- LeVeque, R. J.: Finite Difference Methods, SIAM.
Worked Engineering Example
Problem Statement:
An acoustic wave mode has imaginary eigenvalue $\lambda = i 250\text{ rad/s}$. RK4 has imaginary stability limit $|z_{imag}| \le 2.828$. Calculate maximum stable $\Delta t$.
Step-by-step Solution:
1. $250 \cdot \Delta t \le 2.828 \implies \Delta t_{max} = 2.828 / 250 = 0.01131\text{ s} = 11.31\text{ ms}$.
Final Result:
Maximum stable time step is $\Delta t = \mathbf{11.31\text{ ms}}$.
An acoustic wave mode has imaginary eigenvalue $\lambda = i 250\text{ rad/s}$. RK4 has imaginary stability limit $|z_{imag}| \le 2.828$. Calculate maximum stable $\Delta t$.
Step-by-step Solution:
1. $250 \cdot \Delta t \le 2.828 \implies \Delta t_{max} = 2.828 / 250 = 0.01131\text{ s} = 11.31\text{ ms}$.
Final Result:
Maximum stable time step is $\Delta t = \mathbf{11.31\text{ ms}}$.